msubrn - Mathbox for Mario Carneiro

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Theorem msubrn 35730

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 2737	. . . . . 6 ⊢ (mEx‘𝑇) = (mEx‘𝑇)
5		eqid 2737	. . . . . 6 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇)
6	1, 2, 3, 4, 5	msubffval 35724	. . . . 5 ⊢ (𝑇 ∈ V → 𝑆 = (𝑓 ∈ (𝑅 ↑_pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩)))
7	6	rneqd 5888	. . . 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 35713	. . . . . . . . . 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 6667	. . . . . . . 8 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → Fun (mRSubst‘𝑇))
11	8	ffnd 6664	. . . . . . . . . 10 ⊢ (𝑇 ∈ V → (mRSubst‘𝑇) Fn (𝑅 ↑_pm 𝑉))
12		fnfvelrn 7027	. . . . . . . . . 10 ⊢ (((mRSubst‘𝑇) Fn (𝑅 ↑_pm 𝑉) ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ ran (mRSubst‘𝑇))
13	11, 12	sylan 581	. . . . . . . . 9 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ ran (mRSubst‘𝑇))
14	1, 2, 5	mrsubrn 35714	. . . . . . . . 9 ⊢ ran (mRSubst‘𝑇) = ((mRSubst‘𝑇) “ (𝑅 ↑_m 𝑉))
15	13, 14	eleqtrdi 2847	. . . . . . . 8 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ ((mRSubst‘𝑇) “ (𝑅 ↑_m 𝑉)))
16		fvelima 6900	. . . . . . . 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 585	. . . . . . 7 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → ∃𝑔 ∈ (𝑅 ↑_m 𝑉)((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓))
18		elmapi 8790	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ (𝑅 ↑_m 𝑉) → 𝑔:𝑉⟶𝑅)
19	18	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉)) → 𝑔:𝑉⟶𝑅)
20		ssid 3945	. . . . . . . . . . . 12 ⊢ 𝑉 ⊆ 𝑉
21	1, 2, 3, 4, 5	msubfval 35725	. . . . . . . . . . . 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 587	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉)) → (𝑆‘𝑔) = (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑔)‘(2^nd ‘𝑒))⟩))
23		fvex 6848	. . . . . . . . . . . . . . . 16 ⊢ (mEx‘𝑇) ∈ V
24	23	mptex 7172	. . . . . . . . . . . . . . 15 ⊢ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩) ∈ V
25		eqid 2737	. . . . . . . . . . . . . . 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 6636	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (𝑅 ↑_pm 𝑉) ↦ (𝑒 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩)) Fn (𝑅 ↑_pm 𝑉)
27	6	fneq1d 6586	. . . . . . . . . . . . . 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 8818	. . . . . . . . . . . . 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 7182	. . . . . . . . . . . 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 1374	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ V ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉)) → (𝑆‘𝑔) ∈ (𝑆 “ (𝑅 ↑_m 𝑉)))
35	22, 34	eqeltrrd 2838	. . . . . . . . . 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 716	. . . . . . . . 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 6834	. . . . . . . . . . . 12 ⊢ (((mRSubst‘𝑇)‘𝑔) = ((mRSubst‘𝑇)‘𝑓) → (((mRSubst‘𝑇)‘𝑔)‘(2^nd ‘𝑒)) = (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒)))
38	37	opeq2d 4824	. . . . . . . . . . 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 5180	. . . . . . . . . 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 2822	. . . . . . . . 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 245	. . . . . . . 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 3139	. . . . . . 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 7062	. . . . 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 6671	. . . 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 3957	. . 3 ⊢ (𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅 ↑_m 𝑉)))
47	3	rnfvprc 6829	. . . 4 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 = ∅)
48		0ss 4341	. . . 4 ⊢ ∅ ⊆ (𝑆 “ (𝑅 ↑_m 𝑉))
49	47, 48	eqsstrdi 3967	. . 3 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 ⊆ (𝑆 “ (𝑅 ↑_m 𝑉)))
50	46, 49	pm2.61i 182	. 2 ⊢ ran 𝑆 ⊆ (𝑆 “ (𝑅 ↑_m 𝑉))
51		imassrn 6031	. 2 ⊢ (𝑆 “ (𝑅 ↑_m 𝑉)) ⊆ ran 𝑆
52	50, 51	eqssi 3939	1 ⊢ ran 𝑆 = (𝑆 “ (𝑅 ↑_m 𝑉))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∃wrex 3062 Vcvv 3430 ⊆ wss 3890 ∅c0 4274 ⟨cop 4574 ↦ cmpt 5167 ran crn 5626 “ cima 5628 Fun wfun 6487 Fn wfn 6488 ⟶wf 6489 ‘cfv 6493 (class class class)co 7361 1^st c1st 7934 2^nd c2nd 7935 ↑_m cmap 8767 ↑_pm cpm 8768 mVRcmvar 35662 mRExcmrex 35667 mExcmex 35668 mRSubstcmrsub 35671 mSubstcmsub 35672

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7812 df-1st 7936 df-2nd 7937 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-map 8769 df-pm 8770 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-card 9857 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-nn 12169 df-2 12238 df-n0 12432 df-z 12519 df-uz 12783 df-fz 13456 df-fzo 13603 df-seq 13958 df-hash 14287 df-word 14470 df-concat 14527 df-s1 14553 df-struct 17111 df-sets 17128 df-slot 17146 df-ndx 17158 df-base 17174 df-ress 17195 df-plusg 17227 df-0g 17398 df-gsum 17399 df-mgm 18602 df-sgrp 18681 df-mnd 18697 df-submnd 18746 df-frmd 18811 df-mrex 35687 df-mrsub 35691 df-msub 35692

This theorem is referenced by: msubff1o 35758

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