msubrn - Mathbox for Mario Carneiro

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Theorem msubrn 32889

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 2798	. . . . . 6 ⊢ (mEx‘𝑇) = (mEx‘𝑇)
5		eqid 2798	. . . . . 6 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇)
6	1, 2, 3, 4, 5	msubffval 32883	. . . . 5 ⊢ (𝑇 ∈ V → 𝑆 = (𝑓 ∈ (𝑅 ↑_pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩)))
7	6	rneqd 5772	. . . 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 32872	. . . . . . . . . 10 ⊢ (𝑇 ∈ V → (mRSubst‘𝑇):(𝑅 ↑_pm 𝑉)⟶(𝑅 ↑_m 𝑅))
9	8	adantr 484	. . . . . . . . 9 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → (mRSubst‘𝑇):(𝑅 ↑_pm 𝑉)⟶(𝑅 ↑_m 𝑅))
10	9	ffund 6491	. . . . . . . 8 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → Fun (mRSubst‘𝑇))
11	8	ffnd 6488	. . . . . . . . . 10 ⊢ (𝑇 ∈ V → (mRSubst‘𝑇) Fn (𝑅 ↑_pm 𝑉))
12		fnfvelrn 6825	. . . . . . . . . 10 ⊢ (((mRSubst‘𝑇) Fn (𝑅 ↑_pm 𝑉) ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ ran (mRSubst‘𝑇))
13	11, 12	sylan 583	. . . . . . . . 9 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ ran (mRSubst‘𝑇))
14	1, 2, 5	mrsubrn 32873	. . . . . . . . 9 ⊢ ran (mRSubst‘𝑇) = ((mRSubst‘𝑇) “ (𝑅 ↑_m 𝑉))
15	13, 14	eleqtrdi 2900	. . . . . . . 8 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ ((mRSubst‘𝑇) “ (𝑅 ↑_m 𝑉)))
16		fvelima 6706	. . . . . . . 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 587	. . . . . . 7 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → ∃𝑔 ∈ (𝑅 ↑_m 𝑉)((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓))
18		elmapi 8411	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ (𝑅 ↑_m 𝑉) → 𝑔:𝑉⟶𝑅)
19	18	adantl 485	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉)) → 𝑔:𝑉⟶𝑅)
20		ssid 3937	. . . . . . . . . . . 12 ⊢ 𝑉 ⊆ 𝑉
21	1, 2, 3, 4, 5	msubfval 32884	. . . . . . . . . . . 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 589	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉)) → (𝑆‘𝑔) = (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2^nd ‘𝑒))⟩))
23		fvex 6658	. . . . . . . . . . . . . . . 16 ⊢ (mEx‘𝑇) ∈ V
24	23	mptex 6963	. . . . . . . . . . . . . . 15 ⊢ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩) ∈ V
25		eqid 2798	. . . . . . . . . . . . . . 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 6463	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (𝑅 ↑_pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩)) Fn (𝑅 ↑_pm 𝑉)
27	6	fneq1d 6416	. . . . . . . . . . . . . 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 261	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ V → 𝑆 Fn (𝑅 ↑_pm 𝑉))
29	28	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉)) → 𝑆 Fn (𝑅 ↑_pm 𝑉))
30		mapsspm 8423	. . . . . . . . . . . . 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 488	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉)) → 𝑔 ∈ (𝑅 ↑_m 𝑉))
33		fnfvima 6973	. . . . . . . . . . . 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 1368	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉)) → (𝑆‘𝑔) ∈ (𝑆 “ (𝑅 ↑_m 𝑉)))
35	22, 34	eqeltrrd 2891	. . . . . . . . . 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 6644	. . . . . . . . . . . 12 ⊢ (((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → (((mRSubst‘𝑇)‘𝑔)‘(2^nd ‘𝑒)) = (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒)))
38	37	opeq2d 4772	. . . . . . . . . . 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 5126	. . . . . . . . . 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 2874	. . . . . . . . 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 248	. . . . . . . 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 3243	. . . . . . 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 6856	. . . . 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 6494	. . . 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 3953	. . 3 ⊢ (𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅 ↑_m 𝑉)))
47	3	rnfvprc 6639	. . . 4 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 = ∅)
48		0ss 4304	. . . 4 ⊢ ∅ ⊆ (𝑆 “ (𝑅 ↑_m 𝑉))
49	47, 48	eqsstrdi 3969	. . 3 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅 ↑_m 𝑉)))
50	46, 49	pm2.61i 185	. 2 ⊢ ran 𝑆 ⊆ (𝑆 “ (𝑅 ↑_m 𝑉))
51		imassrn 5907	. 2 ⊢ (𝑆 “ (𝑅 ↑_m 𝑉)) ⊆ ran 𝑆
52	50, 51	eqssi 3931	1 ⊢ ran 𝑆 = (𝑆 “ (𝑅 ↑_m 𝑉))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 Vcvv 3441 ⊆ wss 3881 ∅c0 4243 ⟨cop 4531 ↦ cmpt 5110 ran crn 5520 “ cima 5522 Fun wfun 6318 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 1^st c1st 7669 2^nd c2nd 7670 ↑_m cmap 8389 ↑_pm cpm 8390 mVRcmvar 32821 mRExcmrex 32826 mExcmex 32827 mRSubstcmrsub 32830 mSubstcmsub 32831

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 df-word 13858 df-concat 13914 df-s1 13941 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-0g 16707 df-gsum 16708 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-frmd 18006 df-mrex 32846 df-mrsub 32850 df-msub 32851

This theorem is referenced by: msubff1o 32917

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