msubrn - Mathbox for Mario Carneiro

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Theorem msubrn 35139

Description: Although it is defined for partial mappings of variables, every partial substitution is a substitution on some complete mapping of the variables. (Contributed by Mario Carneiro, 18-Jul-2016.)

**Hypotheses**
Ref	Expression
msubff.v	⊢ 𝑉 = (mVR‘𝑇)
msubff.r	⊢ 𝑅 = (mREx‘𝑇)
msubff.s	⊢ 𝑆 = (mSubst‘𝑇)

**Assertion**
Ref	Expression
msubrn	⊢ ran 𝑆 = (𝑆 “ (𝑅 ↑_m 𝑉))

Proof of Theorem msubrn Dummy variables 𝑒 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		msubff.v	. . . . . 6 ⊢ 𝑉 = (mVR‘𝑇)
2		msubff.r	. . . . . 6 ⊢ 𝑅 = (mREx‘𝑇)
3		msubff.s	. . . . . 6 ⊢ 𝑆 = (mSubst‘𝑇)
4		eqid 2728	. . . . . 6 ⊢ (mEx‘𝑇) = (mEx‘𝑇)
5		eqid 2728	. . . . . 6 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇)
6	1, 2, 3, 4, 5	msubffval 35133	. . . . 5 ⊢ (𝑇 ∈ V → 𝑆 = (𝑓 ∈ (𝑅 ↑_pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩)))
7	6	rneqd 5940	. . . 4 ⊢ (𝑇 ∈ V → ran 𝑆 = ran (𝑓 ∈ (𝑅 ↑_pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩)))
8	1, 2, 5	mrsubff 35122	. . . . . . . . . 10 ⊢ (𝑇 ∈ V → (mRSubst‘𝑇):(𝑅 ↑_pm 𝑉)⟶(𝑅 ↑_m 𝑅))
9	8	adantr 480	. . . . . . . . 9 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → (mRSubst‘𝑇):(𝑅 ↑_pm 𝑉)⟶(𝑅 ↑_m 𝑅))
10	9	ffund 6726	. . . . . . . 8 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → Fun (mRSubst‘𝑇))
11	8	ffnd 6723	. . . . . . . . . 10 ⊢ (𝑇 ∈ V → (mRSubst‘𝑇) Fn (𝑅 ↑_pm 𝑉))
12		fnfvelrn 7090	. . . . . . . . . 10 ⊢ (((mRSubst‘𝑇) Fn (𝑅 ↑_pm 𝑉) ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ ran (mRSubst‘𝑇))
13	11, 12	sylan 579	. . . . . . . . 9 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ ran (mRSubst‘𝑇))
14	1, 2, 5	mrsubrn 35123	. . . . . . . . 9 ⊢ ran (mRSubst‘𝑇) = ((mRSubst‘𝑇) “ (𝑅 ↑_m 𝑉))
15	13, 14	eleqtrdi 2839	. . . . . . . 8 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ ((mRSubst‘𝑇) “ (𝑅 ↑_m 𝑉)))
16		fvelima 6964	. . . . . . . 8 ⊢ ((Fun (mRSubst‘𝑇) ∧ ((mRSubst‘𝑇)‘𝑓) ∈ ((mRSubst‘𝑇) “ (𝑅 ↑_m 𝑉))) → ∃𝑔 ∈ (𝑅 ↑_m 𝑉)((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓))
17	10, 15, 16	syl2anc 583	. . . . . . 7 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → ∃𝑔 ∈ (𝑅 ↑_m 𝑉)((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓))
18		elmapi 8867	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ (𝑅 ↑_m 𝑉) → 𝑔:𝑉⟶𝑅)
19	18	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉)) → 𝑔:𝑉⟶𝑅)
20		ssid 4002	. . . . . . . . . . . 12 ⊢ 𝑉 ⊆ 𝑉
21	1, 2, 3, 4, 5	msubfval 35134	. . . . . . . . . . . 12 ⊢ ((𝑔:𝑉⟶𝑅 ∧ 𝑉 ⊆ 𝑉) → (𝑆‘𝑔) = (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2^nd ‘𝑒))⟩))
22	19, 20, 21	sylancl 585	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉)) → (𝑆‘𝑔) = (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2^nd ‘𝑒))⟩))
23		fvex 6910	. . . . . . . . . . . . . . . 16 ⊢ (mEx‘𝑇) ∈ V
24	23	mptex 7235	. . . . . . . . . . . . . . 15 ⊢ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩) ∈ V
25		eqid 2728	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ (𝑅 ↑_pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩)) = (𝑓 ∈ (𝑅 ↑_pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩))
26	24, 25	fnmpti 6698	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (𝑅 ↑_pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩)) Fn (𝑅 ↑_pm 𝑉)
27	6	fneq1d 6647	. . . . . . . . . . . . . 14 ⊢ (𝑇 ∈ V → (𝑆 Fn (𝑅 ↑_pm 𝑉) ↔ (𝑓 ∈ (𝑅 ↑_pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩)) Fn (𝑅 ↑_pm 𝑉)))
28	26, 27	mpbiri 258	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ V → 𝑆 Fn (𝑅 ↑_pm 𝑉))
29	28	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉)) → 𝑆 Fn (𝑅 ↑_pm 𝑉))
30		mapsspm 8894	. . . . . . . . . . . . 13 ⊢ (𝑅 ↑_m 𝑉) ⊆ (𝑅 ↑_pm 𝑉)
31	30	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉)) → (𝑅 ↑_m 𝑉) ⊆ (𝑅 ↑_pm 𝑉))
32		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉)) → 𝑔 ∈ (𝑅 ↑_m 𝑉))
33		fnfvima 7245	. . . . . . . . . . . 12 ⊢ ((𝑆 Fn (𝑅 ↑_pm 𝑉) ∧ (𝑅 ↑_m 𝑉) ⊆ (𝑅 ↑_pm 𝑉) ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉)) → (𝑆‘𝑔) ∈ (𝑆 “ (𝑅 ↑_m 𝑉)))
34	29, 31, 32, 33	syl3anc 1369	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉)) → (𝑆‘𝑔) ∈ (𝑆 “ (𝑅 ↑_m 𝑉)))
35	22, 34	eqeltrrd 2830	. . . . . . . . . 10 ⊢ ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉)) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2^nd ‘𝑒))⟩) ∈ (𝑆 “ (𝑅 ↑_m 𝑉)))
36	35	adantlr 714	. . . . . . . . 9 ⊢ (((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉)) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2^nd ‘𝑒))⟩) ∈ (𝑆 “ (𝑅 ↑_m 𝑉)))
37		fveq1 6896	. . . . . . . . . . . 12 ⊢ (((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → (((mRSubst‘𝑇)‘𝑔)‘(2^nd ‘𝑒)) = (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒)))
38	37	opeq2d 4881	. . . . . . . . . . 11 ⊢ (((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2^nd ‘𝑒))⟩ = ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩)
39	38	mpteq2dv 5250	. . . . . . . . . 10 ⊢ (((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2^nd ‘𝑒))⟩) = (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩))
40	39	eleq1d 2814	. . . . . . . . 9 ⊢ (((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → ((𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2^nd ‘𝑒))⟩) ∈ (𝑆 “ (𝑅 ↑_m 𝑉)) ↔ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩) ∈ (𝑆 “ (𝑅 ↑_m 𝑉))))
41	36, 40	syl5ibcom 244	. . . . . . . 8 ⊢ (((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉)) → (((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩) ∈ (𝑆 “ (𝑅 ↑_m 𝑉))))
42	41	rexlimdva 3152	. . . . . . 7 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → (∃𝑔 ∈ (𝑅 ↑_m 𝑉)((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩) ∈ (𝑆 “ (𝑅 ↑_m 𝑉))))
43	17, 42	mpd 15	. . . . . 6 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩) ∈ (𝑆 “ (𝑅 ↑_m 𝑉)))
44	43	fmpttd 7125	. . . . 5 ⊢ (𝑇 ∈ V → (𝑓 ∈ (𝑅 ↑_pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩)):(𝑅 ↑_pm 𝑉)⟶(𝑆 “ (𝑅 ↑_m 𝑉)))
45	44	frnd 6730	. . . 4 ⊢ (𝑇 ∈ V → ran (𝑓 ∈ (𝑅 ↑_pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩)) ⊆ (𝑆 “ (𝑅 ↑_m 𝑉)))
46	7, 45	eqsstrd 4018	. . 3 ⊢ (𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅 ↑_m 𝑉)))
47	3	rnfvprc 6891	. . . 4 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 = ∅)
48		0ss 4397	. . . 4 ⊢ ∅ ⊆ (𝑆 “ (𝑅 ↑_m 𝑉))
49	47, 48	eqsstrdi 4034	. . 3 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅 ↑_m 𝑉)))
50	46, 49	pm2.61i 182	. 2 ⊢ ran 𝑆 ⊆ (𝑆 “ (𝑅 ↑_m 𝑉))
51		imassrn 6074	. 2 ⊢ (𝑆 “ (𝑅 ↑_m 𝑉)) ⊆ ran 𝑆
52	50, 51	eqssi 3996	1 ⊢ ran 𝑆 = (𝑆 “ (𝑅 ↑_m 𝑉))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∃wrex 3067 Vcvv 3471 ⊆ wss 3947 ∅c0 4323 ⟨cop 4635 ↦ cmpt 5231 ran crn 5679 “ cima 5681 Fun wfun 6542 Fn wfn 6543 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 1^st c1st 7991 2^nd c2nd 7992 ↑_m cmap 8844 ↑_pm cpm 8845 mVRcmvar 35071 mRExcmrex 35076 mExcmex 35077 mRSubstcmrsub 35080 mSubstcmsub 35081

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-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215

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-nel 3044 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-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-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-map 8846 df-pm 8847 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-fzo 13660 df-seq 13999 df-hash 14322 df-word 14497 df-concat 14553 df-s1 14578 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-0g 17422 df-gsum 17423 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18740 df-frmd 18800 df-mrex 35096 df-mrsub 35100 df-msub 35101

This theorem is referenced by: msubff1o 35167

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