Theorem bnj1006 34596

Description: Technical lemma for bnj69 34646. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1006.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj1006.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1006.3	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj1006.4	⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
bnj1006.5	⊢ (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))
bnj1006.6	⊢ (𝜂 ↔ (𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖)))
bnj1006.7	⊢ (𝜑′ ↔ [𝑝 / 𝑛]𝜑)
bnj1006.8	⊢ (𝜓′ ↔ [𝑝 / 𝑛]𝜓)
bnj1006.9	⊢ (𝜒′ ↔ [𝑝 / 𝑛]𝜒)
bnj1006.10	⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑′)
bnj1006.11	⊢ (𝜓″ ↔ [𝐺 / 𝑓]𝜓′)
bnj1006.12	⊢ (𝜒″ ↔ [𝐺 / 𝑓]𝜒′)
bnj1006.13	⊢ 𝐷 = (ω ∖ {∅})
bnj1006.15	⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
bnj1006.16	⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
bnj1006.28	⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → (𝜒″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝))

Assertion

Ref	Expression
bnj1006	⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → pred(𝑦, 𝐴, 𝑅) ⊆ (𝐺‘suc 𝑖))

Distinct variable groups:   𝐴,𝑓,𝑖,𝑚,𝑛,𝑦   𝐷,𝑓,𝑛   𝑖,𝐺   𝑅,𝑓,𝑖,𝑚,𝑛,𝑦   𝑓,𝑋,𝑛   𝑓,𝑝,𝑖,𝑛

Allowed substitution hints:   𝜑(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜃(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜏(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜂(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐴(𝑧,𝑝)   𝐶(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑦,𝑧,𝑖,𝑚,𝑝)   𝑅(𝑧,𝑝)   𝐺(𝑦,𝑧,𝑓,𝑚,𝑛,𝑝)   𝑋(𝑦,𝑧,𝑖,𝑚,𝑝)   𝜑′(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓′(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒′(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜑″(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓″(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒″(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj1006

Step	Hyp	Ref	Expression
1		bnj1006.6	. . . . 5 ⊢ (𝜂 ↔ (𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖)))
2	1	simprbi 495	. . . 4 ⊢ (𝜂 → 𝑦 ∈ (𝑓‘𝑖))
3	2	bnj708 34392	. . 3 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝑦 ∈ (𝑓‘𝑖))
4		bnj1006.4	. . . . . . . 8 ⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
5		bnj253 34340	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
6	5	simp1bi 1142	. . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)) → (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴))
7	4, 6	sylbi 216	. . . . . . 7 ⊢ (𝜃 → (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴))
8	7	bnj705 34389	. . . . . 6 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴))
9		bnj643 34385	. . . . . . 7 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝜒)
10		bnj1006.5	. . . . . . . . 9 ⊢ (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))
11		3simpc 1147	. . . . . . . . 9 ⊢ ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → (𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))
12	10, 11	sylbi 216	. . . . . . . 8 ⊢ (𝜏 → (𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))
13	12	bnj707 34391	. . . . . . 7 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → (𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))
14		3anass 1092	. . . . . . 7 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ↔ (𝜒 ∧ (𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)))
15	9, 13, 14	sylanbrc 581	. . . . . 6 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))
16		bnj1006.1	. . . . . . 7 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
17		bnj1006.2	. . . . . . 7 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
18		bnj1006.3	. . . . . . 7 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
19		bnj1006.13	. . . . . . 7 ⊢ 𝐷 = (ω ∖ {∅})
20		bnj1006.15	. . . . . . 7 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
21		biid 260	. . . . . . 7 ⊢ ((𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
22		biid 260	. . . . . . 7 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑚 ∈ 𝑛) ↔ (𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑚 ∈ 𝑛))
23	16, 17, 18, 19, 20, 21, 22	bnj969 34582	. . . . . 6 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝐶 ∈ V)
24	8, 15, 23	syl2anc 582	. . . . 5 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝐶 ∈ V)
25	18	bnj1235 34440	. . . . . 6 ⊢ (𝜒 → 𝑓 Fn 𝑛)
26	25	bnj706 34390	. . . . 5 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝑓 Fn 𝑛)
27	10	simp3bi 1144	. . . . . 6 ⊢ (𝜏 → 𝑝 = suc 𝑛)
28	27	bnj707 34391	. . . . 5 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝑝 = suc 𝑛)
29	1	simplbi 496	. . . . . 6 ⊢ (𝜂 → 𝑖 ∈ 𝑛)
30	29	bnj708 34392	. . . . 5 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝑖 ∈ 𝑛)
31	24, 26, 28, 30	bnj951 34411	. . . 4 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → (𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝑖 ∈ 𝑛))
32		bnj1006.16	. . . . 5 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
33	32	bnj945 34409	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝑖 ∈ 𝑛) → (𝐺‘𝑖) = (𝑓‘𝑖))
34	31, 33	syl 17	. . 3 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → (𝐺‘𝑖) = (𝑓‘𝑖))
35	3, 34	eleqtrrd 2831	. 2 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝑦 ∈ (𝐺‘𝑖))
36		bnj1006.28	. . . . 5 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → (𝜒″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝))
37	36	anim1i 613	. . . 4 ⊢ (((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) ∧ 𝑦 ∈ (𝐺‘𝑖)) → ((𝜒″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝) ∧ 𝑦 ∈ (𝐺‘𝑖)))
38		df-bnj17 34323	. . . 4 ⊢ ((𝜒″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑦 ∈ (𝐺‘𝑖)) ↔ ((𝜒″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝) ∧ 𝑦 ∈ (𝐺‘𝑖)))
39	37, 38	sylibr 233	. . 3 ⊢ (((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) ∧ 𝑦 ∈ (𝐺‘𝑖)) → (𝜒″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑦 ∈ (𝐺‘𝑖)))
40		bnj1006.7	. . . 4 ⊢ (𝜑′ ↔ [𝑝 / 𝑛]𝜑)
41		bnj1006.8	. . . 4 ⊢ (𝜓′ ↔ [𝑝 / 𝑛]𝜓)
42		bnj1006.9	. . . 4 ⊢ (𝜒′ ↔ [𝑝 / 𝑛]𝜒)
43		bnj1006.10	. . . 4 ⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑′)
44		bnj1006.11	. . . 4 ⊢ (𝜓″ ↔ [𝐺 / 𝑓]𝜓′)
45		bnj1006.12	. . . 4 ⊢ (𝜒″ ↔ [𝐺 / 𝑓]𝜒′)
46	16, 17, 18, 40, 41, 42, 43, 44, 45, 20, 32	bnj999 34594	. . 3 ⊢ ((𝜒″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑦 ∈ (𝐺‘𝑖)) → pred(𝑦, 𝐴, 𝑅) ⊆ (𝐺‘suc 𝑖))
47	39, 46	syl 17	. 2 ⊢ (((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) ∧ 𝑦 ∈ (𝐺‘𝑖)) → pred(𝑦, 𝐴, 𝑅) ⊆ (𝐺‘suc 𝑖))
48	35, 47	mpdan 685	1 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → pred(𝑦, 𝐴, 𝑅) ⊆ (𝐺‘suc 𝑖))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3057  Vcvv 3471  [wsbc 3776   ∖ cdif 3944   ∪ cun 3945   ⊆ wss 3947  ∅c0 4324  {csn 4630  ⟨cop 4636  ∪ ciun 4998  suc csuc 6374   Fn wfn 6546  ‘cfv 6551  ωcom 7874   ∧ w-bnj17 34322   predc-bnj14 34324   FrSe w-bnj15 34328   trClc-bnj18 34330

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pr 5431  ax-un 7744  ax-reg 9621

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-res 5692  df-ord 6375  df-on 6376  df-lim 6377  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-fv 6559  df-om 7875  df-bnj17 34323  df-bnj14 34325  df-bnj13 34327  df-bnj15 34329

This theorem is referenced by:  bnj1020  34601

Otomatik - 173.255.232.114 CloudFlare DNS Türk Telekom DNS Google DNS Open DNS

OSZAR »