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Theorem cnfcom3 9727

Description: Any infinite ordinal 𝐵 is equinumerous to a power of ω. (We are being careful here to show explicit bijections rather than simple equinumerosity because we want a uniform construction for cnfcom3c 9729.) (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 4-Jul-2019.)

**Hypotheses**
Ref	Expression
cnfcom.s	⊢ 𝑆 = dom (ω CNF 𝐴)
cnfcom.a	⊢ (𝜑 → 𝐴 ∈ On)
cnfcom.b	⊢ (𝜑 → 𝐵 ∈ (ω ↑_o 𝐴))
cnfcom.f	⊢ 𝐹 = (^◡(ω CNF 𝐴)‘𝐵)
cnfcom.g	⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))
cnfcom.h	⊢ 𝐻 = seq_ω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +_o 𝑧)), ∅)
cnfcom.t	⊢ 𝑇 = seq_ω((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)
cnfcom.m	⊢ 𝑀 = ((ω ↑_o (𝐺‘𝑘)) ·_o (𝐹‘(𝐺‘𝑘)))
cnfcom.k	⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +_o 𝑥)) ∪ ^◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +_o 𝑥)))
cnfcom.w	⊢ 𝑊 = (𝐺‘∪ dom 𝐺)
cnfcom3.1	⊢ (𝜑 → ω ⊆ 𝐵)
cnfcom.x	⊢ 𝑋 = (𝑢 ∈ (𝐹‘𝑊), 𝑣 ∈ (ω ↑_o 𝑊) ↦ (((𝐹‘𝑊) ·_o 𝑣) +_o 𝑢))
cnfcom.y	⊢ 𝑌 = (𝑢 ∈ (𝐹‘𝑊), 𝑣 ∈ (ω ↑_o 𝑊) ↦ (((ω ↑_o 𝑊) ·_o 𝑢) +_o 𝑣))
cnfcom.n	⊢ 𝑁 = ((𝑋 ∘ ^◡𝑌) ∘ (𝑇‘dom 𝐺))

**Assertion**
Ref	Expression
cnfcom3	⊢ (𝜑 → 𝑁:𝐵–1-1-onto→(ω ↑_o 𝑊))

Distinct variable groups: 𝑥,𝑘,𝑧,𝐴 𝑢,𝑘,𝑣,𝑥,𝑧 𝑥,𝑀 𝜑,𝑢,𝑣 𝑓,𝑘,𝑢,𝑣,𝑥,𝑧,𝐹 𝑢,𝐾,𝑣 𝑢,𝑇,𝑣,𝑧 𝑢,𝑊,𝑣,𝑥 𝑓,𝐺,𝑘,𝑢,𝑣,𝑥,𝑧 𝑓,𝐻,𝑢,𝑣,𝑥 𝑆,𝑘,𝑧 𝜑,𝑘,𝑥,𝑧 Allowed substitution hints: 𝜑(𝑓) 𝐴(𝑣,𝑢,𝑓) 𝐵(𝑥,𝑧,𝑣,𝑢,𝑓,𝑘) 𝑆(𝑥,𝑣,𝑢,𝑓) 𝑇(𝑥,𝑓,𝑘) 𝐻(𝑧,𝑘) 𝐾(𝑥,𝑧,𝑓,𝑘) 𝑀(𝑧,𝑣,𝑢,𝑓,𝑘) 𝑁(𝑥,𝑧,𝑣,𝑢,𝑓,𝑘) 𝑊(𝑧,𝑓,𝑘) 𝑋(𝑥,𝑧,𝑣,𝑢,𝑓,𝑘) 𝑌(𝑥,𝑧,𝑣,𝑢,𝑓,𝑘)

**Proof of Theorem cnfcom3**
Step	Hyp	Ref	Expression
1		omelon 9669	. . . . . 6 ⊢ ω ∈ On
2		cnfcom.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ On)
3		suppssdm 8181	. . . . . . . . 9 ⊢ (𝐹 supp ∅) ⊆ dom 𝐹
4		cnfcom.f	. . . . . . . . . . . 12 ⊢ 𝐹 = (^◡(ω CNF 𝐴)‘𝐵)
5		cnfcom.s	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = dom (ω CNF 𝐴)
6	1	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ω ∈ On)
7	5, 6, 2	cantnff1o 9719	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ω CNF 𝐴):𝑆–1-1-onto→(ω ↑_o 𝐴))
8		f1ocnv 6851	. . . . . . . . . . . . . 14 ⊢ ((ω CNF 𝐴):𝑆–1-1-onto→(ω ↑_o 𝐴) → ^◡(ω CNF 𝐴):(ω ↑_o 𝐴)–1-1-onto→𝑆)
9		f1of 6839	. . . . . . . . . . . . . 14 ⊢ (^◡(ω CNF 𝐴):(ω ↑_o 𝐴)–1-1-onto→𝑆 → ^◡(ω CNF 𝐴):(ω ↑_o 𝐴)⟶𝑆)
10	7, 8, 9	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → ^◡(ω CNF 𝐴):(ω ↑_o 𝐴)⟶𝑆)
11		cnfcom.b	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ (ω ↑_o 𝐴))
12	10, 11	ffvelcdmd 7095	. . . . . . . . . . . 12 ⊢ (𝜑 → (^◡(ω CNF 𝐴)‘𝐵) ∈ 𝑆)
13	4, 12	eqeltrid 2833	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ 𝑆)
14	5, 6, 2	cantnfs 9689	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅)))
15	13, 14	mpbid 231	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅))
16	15	simpld 494	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝐴⟶ω)
17	3, 16	fssdm 6742	. . . . . . . 8 ⊢ (𝜑 → (𝐹 supp ∅) ⊆ 𝐴)
18		cnfcom.w	. . . . . . . . 9 ⊢ 𝑊 = (𝐺‘∪ dom 𝐺)
19		ovex 7453	. . . . . . . . . . . . . . 15 ⊢ (𝐹 supp ∅) ∈ V
20		cnfcom.g	. . . . . . . . . . . . . . . 16 ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))
21	20	oion 9559	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 supp ∅) ∈ V → dom 𝐺 ∈ On)
22	19, 21	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ dom 𝐺 ∈ On
23	22	elexi 3491	. . . . . . . . . . . . 13 ⊢ dom 𝐺 ∈ V
24	23	uniex 7746	. . . . . . . . . . . 12 ⊢ ∪ dom 𝐺 ∈ V
25	24	sucid 6451	. . . . . . . . . . 11 ⊢ ∪ dom 𝐺 ∈ suc ∪ dom 𝐺
26		cnfcom.h	. . . . . . . . . . . 12 ⊢ 𝐻 = seq_ω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +_o 𝑧)), ∅)
27		cnfcom.t	. . . . . . . . . . . 12 ⊢ 𝑇 = seq_ω((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)
28		cnfcom.m	. . . . . . . . . . . 12 ⊢ 𝑀 = ((ω ↑_o (𝐺‘𝑘)) ·_o (𝐹‘(𝐺‘𝑘)))
29		cnfcom.k	. . . . . . . . . . . 12 ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +_o 𝑥)) ∪ ^◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +_o 𝑥)))
30		cnfcom3.1	. . . . . . . . . . . . 13 ⊢ (𝜑 → ω ⊆ 𝐵)
31		peano1 7894	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ ω
32	31	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∅ ∈ ω)
33	30, 32	sseldd 3981	. . . . . . . . . . . 12 ⊢ (𝜑 → ∅ ∈ 𝐵)
34	5, 2, 11, 4, 20, 26, 27, 28, 29, 18, 33	cnfcom2lem 9724	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐺 = suc ∪ dom 𝐺)
35	25, 34	eleqtrrid 2836	. . . . . . . . . 10 ⊢ (𝜑 → ∪ dom 𝐺 ∈ dom 𝐺)
36	20	oif 9553	. . . . . . . . . . 11 ⊢ 𝐺:dom 𝐺⟶(𝐹 supp ∅)
37	36	ffvelcdmi 7093	. . . . . . . . . 10 ⊢ (∪ dom 𝐺 ∈ dom 𝐺 → (𝐺‘∪ dom 𝐺) ∈ (𝐹 supp ∅))
38	35, 37	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐺‘∪ dom 𝐺) ∈ (𝐹 supp ∅))
39	18, 38	eqeltrid 2833	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ (𝐹 supp ∅))
40	17, 39	sseldd 3981	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ 𝐴)
41		onelon 6394	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑊 ∈ 𝐴) → 𝑊 ∈ On)
42	2, 40, 41	syl2anc 583	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ On)
43		oecl 8557	. . . . . 6 ⊢ ((ω ∈ On ∧ 𝑊 ∈ On) → (ω ↑_o 𝑊) ∈ On)
44	1, 42, 43	sylancr 586	. . . . 5 ⊢ (𝜑 → (ω ↑_o 𝑊) ∈ On)
45	16, 40	ffvelcdmd 7095	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑊) ∈ ω)
46		nnon 7876	. . . . . 6 ⊢ ((𝐹‘𝑊) ∈ ω → (𝐹‘𝑊) ∈ On)
47	45, 46	syl 17	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑊) ∈ On)
48		cnfcom.y	. . . . . 6 ⊢ 𝑌 = (𝑢 ∈ (𝐹‘𝑊), 𝑣 ∈ (ω ↑_o 𝑊) ↦ (((ω ↑_o 𝑊) ·_o 𝑢) +_o 𝑣))
49		cnfcom.x	. . . . . 6 ⊢ 𝑋 = (𝑢 ∈ (𝐹‘𝑊), 𝑣 ∈ (ω ↑_o 𝑊) ↦ (((𝐹‘𝑊) ·_o 𝑣) +_o 𝑢))
50	48, 49	omf1o 9099	. . . . 5 ⊢ (((ω ↑_o 𝑊) ∈ On ∧ (𝐹‘𝑊) ∈ On) → (𝑋 ∘ ^◡𝑌):((ω ↑_o 𝑊) ·_o (𝐹‘𝑊))–1-1-onto→((𝐹‘𝑊) ·_o (ω ↑_o 𝑊)))
51	44, 47, 50	syl2anc 583	. . . 4 ⊢ (𝜑 → (𝑋 ∘ ^◡𝑌):((ω ↑_o 𝑊) ·_o (𝐹‘𝑊))–1-1-onto→((𝐹‘𝑊) ·_o (ω ↑_o 𝑊)))
52	16	ffnd 6723	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 Fn 𝐴)
53		0ex 5307	. . . . . . . . . . 11 ⊢ ∅ ∈ V
54	53	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ∅ ∈ V)
55		elsuppfn 8175	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ On ∧ ∅ ∈ V) → (𝑊 ∈ (𝐹 supp ∅) ↔ (𝑊 ∈ 𝐴 ∧ (𝐹‘𝑊) ≠ ∅)))
56	52, 2, 54, 55	syl3anc 1369	. . . . . . . . 9 ⊢ (𝜑 → (𝑊 ∈ (𝐹 supp ∅) ↔ (𝑊 ∈ 𝐴 ∧ (𝐹‘𝑊) ≠ ∅)))
57		simpr 484	. . . . . . . . 9 ⊢ ((𝑊 ∈ 𝐴 ∧ (𝐹‘𝑊) ≠ ∅) → (𝐹‘𝑊) ≠ ∅)
58	56, 57	biimtrdi 252	. . . . . . . 8 ⊢ (𝜑 → (𝑊 ∈ (𝐹 supp ∅) → (𝐹‘𝑊) ≠ ∅))
59	39, 58	mpd 15	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝑊) ≠ ∅)
60		on0eln0 6425	. . . . . . . 8 ⊢ ((𝐹‘𝑊) ∈ On → (∅ ∈ (𝐹‘𝑊) ↔ (𝐹‘𝑊) ≠ ∅))
61	45, 46, 60	3syl 18	. . . . . . 7 ⊢ (𝜑 → (∅ ∈ (𝐹‘𝑊) ↔ (𝐹‘𝑊) ≠ ∅))
62	59, 61	mpbird 257	. . . . . 6 ⊢ (𝜑 → ∅ ∈ (𝐹‘𝑊))
63	5, 2, 11, 4, 20, 26, 27, 28, 29, 18, 30	cnfcom3lem 9726	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ (On ∖ 1_o))
64		ondif1 8521	. . . . . . . 8 ⊢ (𝑊 ∈ (On ∖ 1_o) ↔ (𝑊 ∈ On ∧ ∅ ∈ 𝑊))
65	64	simprbi 496	. . . . . . 7 ⊢ (𝑊 ∈ (On ∖ 1_o) → ∅ ∈ 𝑊)
66	63, 65	syl 17	. . . . . 6 ⊢ (𝜑 → ∅ ∈ 𝑊)
67		omabs 8671	. . . . . 6 ⊢ ((((𝐹‘𝑊) ∈ ω ∧ ∅ ∈ (𝐹‘𝑊)) ∧ (𝑊 ∈ On ∧ ∅ ∈ 𝑊)) → ((𝐹‘𝑊) ·_o (ω ↑_o 𝑊)) = (ω ↑_o 𝑊))
68	45, 62, 42, 66, 67	syl22anc 838	. . . . 5 ⊢ (𝜑 → ((𝐹‘𝑊) ·_o (ω ↑_o 𝑊)) = (ω ↑_o 𝑊))
69	68	f1oeq3d 6836	. . . 4 ⊢ (𝜑 → ((𝑋 ∘ ^◡𝑌):((ω ↑_o 𝑊) ·_o (𝐹‘𝑊))–1-1-onto→((𝐹‘𝑊) ·_o (ω ↑_o 𝑊)) ↔ (𝑋 ∘ ^◡𝑌):((ω ↑_o 𝑊) ·_o (𝐹‘𝑊))–1-1-onto→(ω ↑_o 𝑊)))
70	51, 69	mpbid 231	. . 3 ⊢ (𝜑 → (𝑋 ∘ ^◡𝑌):((ω ↑_o 𝑊) ·_o (𝐹‘𝑊))–1-1-onto→(ω ↑_o 𝑊))
71	5, 2, 11, 4, 20, 26, 27, 28, 29, 18, 33	cnfcom2 9725	. . 3 ⊢ (𝜑 → (𝑇‘dom 𝐺):𝐵–1-1-onto→((ω ↑_o 𝑊) ·_o (𝐹‘𝑊)))
72		f1oco 6862	. . 3 ⊢ (((𝑋 ∘ ^◡𝑌):((ω ↑_o 𝑊) ·_o (𝐹‘𝑊))–1-1-onto→(ω ↑_o 𝑊) ∧ (𝑇‘dom 𝐺):𝐵–1-1-onto→((ω ↑_o 𝑊) ·_o (𝐹‘𝑊))) → ((𝑋 ∘ ^◡𝑌) ∘ (𝑇‘dom 𝐺)):𝐵–1-1-onto→(ω ↑_o 𝑊))
73	70, 71, 72	syl2anc 583	. 2 ⊢ (𝜑 → ((𝑋 ∘ ^◡𝑌) ∘ (𝑇‘dom 𝐺)):𝐵–1-1-onto→(ω ↑_o 𝑊))
74		cnfcom.n	. . 3 ⊢ 𝑁 = ((𝑋 ∘ ^◡𝑌) ∘ (𝑇‘dom 𝐺))
75		f1oeq1 6827	. . 3 ⊢ (𝑁 = ((𝑋 ∘ ^◡𝑌) ∘ (𝑇‘dom 𝐺)) → (𝑁:𝐵–1-1-onto→(ω ↑_o 𝑊) ↔ ((𝑋 ∘ ^◡𝑌) ∘ (𝑇‘dom 𝐺)):𝐵–1-1-onto→(ω ↑_o 𝑊)))
76	74, 75	ax-mp 5	. 2 ⊢ (𝑁:𝐵–1-1-onto→(ω ↑_o 𝑊) ↔ ((𝑋 ∘ ^◡𝑌) ∘ (𝑇‘dom 𝐺)):𝐵–1-1-onto→(ω ↑_o 𝑊))
77	73, 76	sylibr 233	1 ⊢ (𝜑 → 𝑁:𝐵–1-1-onto→(ω ↑_o 𝑊))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 Vcvv 3471 ∖ cdif 3944 ∪ cun 3945 ⊆ wss 3947 ∅c0 4323 ∪ cuni 4908 class class class wbr 5148 ↦ cmpt 5231 E cep 5581 ^◡ccnv 5677 dom cdm 5678 ∘ ccom 5682 Oncon0 6369 suc csuc 6371 Fn wfn 6543 ⟶wf 6544 –1-1-onto→wf1o 6547 ‘cfv 6548 (class class class)co 7420 ∈ cmpo 7422 ωcom 7870 supp csupp 8165 seq_ωcseqom 8467 1_oc1o 8479 +_o coa 8483 ·_o comu 8484 ↑_o coe 8485 finSupp cfsupp 9385 OrdIsocoi 9532 CNF ccnf 9684

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9664

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-seqom 8468 df-1o 8486 df-2o 8487 df-oadd 8490 df-omul 8491 df-oexp 8492 df-er 8724 df-map 8846 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-fsupp 9386 df-oi 9533 df-cnf 9685

This theorem is referenced by: cnfcom3clem 9728

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