mrsubcn - Mathbox for Mario Carneiro

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Theorem mrsubcn 34113

Description: A substitution does not change the value of constant substrings. (Contributed by Mario Carneiro, 18-Jul-2016.)

**Hypotheses**
Ref	Expression
mrsubccat.s	⊢ 𝑆 = (mRSubst‘𝑇)
mrsubccat.r	⊢ 𝑅 = (mREx‘𝑇)
mrsubcn.v	⊢ 𝑉 = (mVR‘𝑇)
mrsubcn.c	⊢ 𝐶 = (mCN‘𝑇)

**Assertion**
Ref	Expression
mrsubcn	⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝑋 ∈ (𝐶 ∖ 𝑉)) → (𝐹‘⟨“𝑋”⟩) = ⟨“𝑋”⟩)

Proof of Theorem mrsubcn Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0i 4293	. . . . 5 ⊢ (𝐹 ∈ ran 𝑆 → ¬ ran 𝑆 = ∅)
2		mrsubccat.s	. . . . . 6 ⊢ 𝑆 = (mRSubst‘𝑇)
3	2	rnfvprc 6836	. . . . 5 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 = ∅)
4	1, 3	nsyl2 141	. . . 4 ⊢ (𝐹 ∈ ran 𝑆 → 𝑇 ∈ V)
5		mrsubcn.v	. . . . 5 ⊢ 𝑉 = (mVR‘𝑇)
6		mrsubccat.r	. . . . 5 ⊢ 𝑅 = (mREx‘𝑇)
7	5, 6, 2	mrsubff 34106	. . . 4 ⊢ (𝑇 ∈ V → 𝑆:(𝑅 ↑_pm 𝑉)⟶(𝑅 ↑_m 𝑅))
8		ffun 6671	. . . 4 ⊢ (𝑆:(𝑅 ↑_pm 𝑉)⟶(𝑅 ↑_m 𝑅) → Fun 𝑆)
9	4, 7, 8	3syl 18	. . 3 ⊢ (𝐹 ∈ ran 𝑆 → Fun 𝑆)
10	5, 6, 2	mrsubrn 34107	. . . . 5 ⊢ ran 𝑆 = (𝑆 “ (𝑅 ↑_m 𝑉))
11	10	eleq2i 2829	. . . 4 ⊢ (𝐹 ∈ ran 𝑆 ↔ 𝐹 ∈ (𝑆 “ (𝑅 ↑_m 𝑉)))
12	11	biimpi 215	. . 3 ⊢ (𝐹 ∈ ran 𝑆 → 𝐹 ∈ (𝑆 “ (𝑅 ↑_m 𝑉)))
13		fvelima 6908	. . 3 ⊢ ((Fun 𝑆 ∧ 𝐹 ∈ (𝑆 “ (𝑅 ↑_m 𝑉))) → ∃𝑓 ∈ (𝑅 ↑_m 𝑉)(𝑆‘𝑓) = 𝐹)
14	9, 12, 13	syl2anc 584	. 2 ⊢ (𝐹 ∈ ran 𝑆 → ∃𝑓 ∈ (𝑅 ↑_m 𝑉)(𝑆‘𝑓) = 𝐹)
15		elmapi 8787	. . . . . . 7 ⊢ (𝑓 ∈ (𝑅 ↑_m 𝑉) → 𝑓:𝑉⟶𝑅)
16	15	adantl 482	. . . . . 6 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑_m 𝑉)) → 𝑓:𝑉⟶𝑅)
17		ssidd 3967	. . . . . 6 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑_m 𝑉)) → 𝑉 ⊆ 𝑉)
18		eldifi 4086	. . . . . . . 8 ⊢ (𝑋 ∈ (𝐶 ∖ 𝑉) → 𝑋 ∈ 𝐶)
19		elun1 4136	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐶 → 𝑋 ∈ (𝐶 ∪ 𝑉))
20	18, 19	syl 17	. . . . . . 7 ⊢ (𝑋 ∈ (𝐶 ∖ 𝑉) → 𝑋 ∈ (𝐶 ∪ 𝑉))
21	20	adantr 481	. . . . . 6 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑_m 𝑉)) → 𝑋 ∈ (𝐶 ∪ 𝑉))
22		mrsubcn.c	. . . . . . 7 ⊢ 𝐶 = (mCN‘𝑇)
23	22, 5, 6, 2	mrsubcv 34104	. . . . . 6 ⊢ ((𝑓:𝑉⟶𝑅 ∧ 𝑉 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → ((𝑆‘𝑓)‘⟨“𝑋”⟩) = if(𝑋 ∈ 𝑉, (𝑓‘𝑋), ⟨“𝑋”⟩))
24	16, 17, 21, 23	syl3anc 1371	. . . . 5 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑_m 𝑉)) → ((𝑆‘𝑓)‘⟨“𝑋”⟩) = if(𝑋 ∈ 𝑉, (𝑓‘𝑋), ⟨“𝑋”⟩))
25		eldifn 4087	. . . . . . 7 ⊢ (𝑋 ∈ (𝐶 ∖ 𝑉) → ¬ 𝑋 ∈ 𝑉)
26	25	adantr 481	. . . . . 6 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑_m 𝑉)) → ¬ 𝑋 ∈ 𝑉)
27	26	iffalsed 4497	. . . . 5 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑_m 𝑉)) → if(𝑋 ∈ 𝑉, (𝑓‘𝑋), ⟨“𝑋”⟩) = ⟨“𝑋”⟩)
28	24, 27	eqtrd 2776	. . . 4 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑_m 𝑉)) → ((𝑆‘𝑓)‘⟨“𝑋”⟩) = ⟨“𝑋”⟩)
29		fveq1 6841	. . . . 5 ⊢ ((𝑆‘𝑓) = 𝐹 → ((𝑆‘𝑓)‘⟨“𝑋”⟩) = (𝐹‘⟨“𝑋”⟩))
30	29	eqeq1d 2738	. . . 4 ⊢ ((𝑆‘𝑓) = 𝐹 → (((𝑆‘𝑓)‘⟨“𝑋”⟩) = ⟨“𝑋”⟩ ↔ (𝐹‘⟨“𝑋”⟩) = ⟨“𝑋”⟩))
31	28, 30	syl5ibcom 244	. . 3 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑_m 𝑉)) → ((𝑆‘𝑓) = 𝐹 → (𝐹‘⟨“𝑋”⟩) = ⟨“𝑋”⟩))
32	31	rexlimdva 3152	. 2 ⊢ (𝑋 ∈ (𝐶 ∖ 𝑉) → (∃𝑓 ∈ (𝑅 ↑_m 𝑉)(𝑆‘𝑓) = 𝐹 → (𝐹‘⟨“𝑋”⟩) = ⟨“𝑋”⟩))
33	14, 32	mpan9 507	1 ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝑋 ∈ (𝐶 ∖ 𝑉)) → (𝐹‘⟨“𝑋”⟩) = ⟨“𝑋”⟩)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∃wrex 3073 Vcvv 3445 ∖ cdif 3907 ∪ cun 3908 ⊆ wss 3910 ∅c0 4282 ifcif 4486 ran crn 5634 “ cima 5636 Fun wfun 6490 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ↑_m cmap 8765 ↑_pm cpm 8766 ⟨“cs1 14483 mCNcmcn 34054 mVRcmvar 34055 mRExcmrex 34060 mRSubstcmrsub 34064

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-pm 8768 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 df-fzo 13568 df-seq 13907 df-hash 14231 df-word 14403 df-concat 14459 df-s1 14484 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-0g 17323 df-gsum 17324 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-frmd 18659 df-mrex 34080 df-mrsub 34084

This theorem is referenced by: elmrsubrn 34114 mrsubco 34115 mrsubvrs 34116

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