mrsubff - Mathbox for Mario Carneiro

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Theorem mrsubff 34801

Description: A substitution is a function from 𝑅 to 𝑅. (Contributed by Mario Carneiro, 18-Jul-2016.)

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

**Assertion**
Ref	Expression
mrsubff	⊢ (𝑇 ∈ 𝑊 → 𝑆:(𝑅 ↑_pm 𝑉)⟶(𝑅 ↑_m 𝑅))

Proof of Theorem mrsubff Dummy variables 𝑒 𝑓 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6903	. . . . . . . . 9 ⊢ (mCN‘𝑇) ∈ V
2		mrsubvr.v	. . . . . . . . . 10 ⊢ 𝑉 = (mVR‘𝑇)
3	2	fvexi 6904	. . . . . . . . 9 ⊢ 𝑉 ∈ V
4	1, 3	unex 7735	. . . . . . . 8 ⊢ ((mCN‘𝑇) ∪ 𝑉) ∈ V
5		eqid 2730	. . . . . . . . 9 ⊢ (freeMnd‘((mCN‘𝑇) ∪ 𝑉)) = (freeMnd‘((mCN‘𝑇) ∪ 𝑉))
6	5	frmdmnd 18776	. . . . . . . 8 ⊢ (((mCN‘𝑇) ∪ 𝑉) ∈ V → (freeMnd‘((mCN‘𝑇) ∪ 𝑉)) ∈ Mnd)
7	4, 6	mp1i 13	. . . . . . 7 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) ∧ 𝑒 ∈ 𝑅) → (freeMnd‘((mCN‘𝑇) ∪ 𝑉)) ∈ Mnd)
8		simpr 483	. . . . . . . . 9 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) ∧ 𝑒 ∈ 𝑅) → 𝑒 ∈ 𝑅)
9		eqid 2730	. . . . . . . . . . 11 ⊢ (mCN‘𝑇) = (mCN‘𝑇)
10		mrsubvr.r	. . . . . . . . . . 11 ⊢ 𝑅 = (mREx‘𝑇)
11	9, 2, 10	mrexval 34790	. . . . . . . . . 10 ⊢ (𝑇 ∈ 𝑊 → 𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
12	11	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) ∧ 𝑒 ∈ 𝑅) → 𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
13	8, 12	eleqtrd 2833	. . . . . . . 8 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) ∧ 𝑒 ∈ 𝑅) → 𝑒 ∈ Word ((mCN‘𝑇) ∪ 𝑉))
14		elpmi 8842	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (𝑅 ↑_pm 𝑉) → (𝑓:dom 𝑓⟶𝑅 ∧ dom 𝑓 ⊆ 𝑉))
15	14	simpld 493	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (𝑅 ↑_pm 𝑉) → 𝑓:dom 𝑓⟶𝑅)
16	15	ad3antlr 727	. . . . . . . . . . . 12 ⊢ ((((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) ∧ 𝑒 ∈ 𝑅) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) → 𝑓:dom 𝑓⟶𝑅)
17	16	ffvelcdmda 7085	. . . . . . . . . . 11 ⊢ (((((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) ∧ 𝑒 ∈ 𝑅) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) ∧ 𝑣 ∈ dom 𝑓) → (𝑓‘𝑣) ∈ 𝑅)
18	12	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) ∧ 𝑒 ∈ 𝑅) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) ∧ 𝑣 ∈ dom 𝑓) → 𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
19	17, 18	eleqtrd 2833	. . . . . . . . . 10 ⊢ (((((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) ∧ 𝑒 ∈ 𝑅) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) ∧ 𝑣 ∈ dom 𝑓) → (𝑓‘𝑣) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
20		simplr 765	. . . . . . . . . . 11 ⊢ (((((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) ∧ 𝑒 ∈ 𝑅) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) ∧ ¬ 𝑣 ∈ dom 𝑓) → 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉))
21	20	s1cld 14557	. . . . . . . . . 10 ⊢ (((((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) ∧ 𝑒 ∈ 𝑅) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) ∧ ¬ 𝑣 ∈ dom 𝑓) → ⟨“𝑣”⟩ ∈ Word ((mCN‘𝑇) ∪ 𝑉))
22	19, 21	ifclda 4562	. . . . . . . . 9 ⊢ ((((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) ∧ 𝑒 ∈ 𝑅) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) → if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
23	22	fmpttd 7115	. . . . . . . 8 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) ∧ 𝑒 ∈ 𝑅) → (𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)):((mCN‘𝑇) ∪ 𝑉)⟶Word ((mCN‘𝑇) ∪ 𝑉))
24		wrdco 14786	. . . . . . . 8 ⊢ ((𝑒 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ (𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)):((mCN‘𝑇) ∪ 𝑉)⟶Word ((mCN‘𝑇) ∪ 𝑉)) → ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒) ∈ Word Word ((mCN‘𝑇) ∪ 𝑉))
25	13, 23, 24	syl2anc 582	. . . . . . 7 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) ∧ 𝑒 ∈ 𝑅) → ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒) ∈ Word Word ((mCN‘𝑇) ∪ 𝑉))
26		eqid 2730	. . . . . . . . . . 11 ⊢ (Base‘(freeMnd‘((mCN‘𝑇) ∪ 𝑉))) = (Base‘(freeMnd‘((mCN‘𝑇) ∪ 𝑉)))
27	5, 26	frmdbas 18769	. . . . . . . . . 10 ⊢ (((mCN‘𝑇) ∪ 𝑉) ∈ V → (Base‘(freeMnd‘((mCN‘𝑇) ∪ 𝑉))) = Word ((mCN‘𝑇) ∪ 𝑉))
28	4, 27	ax-mp 5	. . . . . . . . 9 ⊢ (Base‘(freeMnd‘((mCN‘𝑇) ∪ 𝑉))) = Word ((mCN‘𝑇) ∪ 𝑉)
29	28	eqcomi 2739	. . . . . . . 8 ⊢ Word ((mCN‘𝑇) ∪ 𝑉) = (Base‘(freeMnd‘((mCN‘𝑇) ∪ 𝑉)))
30	29	gsumwcl 18756	. . . . . . 7 ⊢ (((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) ∈ Mnd ∧ ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒) ∈ Word Word ((mCN‘𝑇) ∪ 𝑉)) → ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
31	7, 25, 30	syl2anc 582	. . . . . 6 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) ∧ 𝑒 ∈ 𝑅) → ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
32	31, 12	eleqtrrd 2834	. . . . 5 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) ∧ 𝑒 ∈ 𝑅) → ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)) ∈ 𝑅)
33	32	fmpttd 7115	. . . 4 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → (𝑒 ∈ 𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))):𝑅⟶𝑅)
34	10	fvexi 6904	. . . . 5 ⊢ 𝑅 ∈ V
35	34, 34	elmap 8867	. . . 4 ⊢ ((𝑒 ∈ 𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))) ∈ (𝑅 ↑_m 𝑅) ↔ (𝑒 ∈ 𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))):𝑅⟶𝑅)
36	33, 35	sylibr 233	. . 3 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → (𝑒 ∈ 𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))) ∈ (𝑅 ↑_m 𝑅))
37	36	fmpttd 7115	. 2 ⊢ (𝑇 ∈ 𝑊 → (𝑓 ∈ (𝑅 ↑_pm 𝑉) ↦ (𝑒 ∈ 𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))):(𝑅 ↑_pm 𝑉)⟶(𝑅 ↑_m 𝑅))
38		mrsubvr.s	. . . 4 ⊢ 𝑆 = (mRSubst‘𝑇)
39	9, 2, 10, 38, 5	mrsubffval 34796	. . 3 ⊢ (𝑇 ∈ 𝑊 → 𝑆 = (𝑓 ∈ (𝑅 ↑_pm 𝑉) ↦ (𝑒 ∈ 𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))))
40	39	feq1d 6701	. 2 ⊢ (𝑇 ∈ 𝑊 → (𝑆:(𝑅 ↑_pm 𝑉)⟶(𝑅 ↑_m 𝑅) ↔ (𝑓 ∈ (𝑅 ↑_pm 𝑉) ↦ (𝑒 ∈ 𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))):(𝑅 ↑_pm 𝑉)⟶(𝑅 ↑_m 𝑅)))
41	37, 40	mpbird 256	1 ⊢ (𝑇 ∈ 𝑊 → 𝑆:(𝑅 ↑_pm 𝑉)⟶(𝑅 ↑_m 𝑅))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 Vcvv 3472 ∪ cun 3945 ⊆ wss 3947 ifcif 4527 ↦ cmpt 5230 dom cdm 5675 ∘ ccom 5679 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 ↑_m cmap 8822 ↑_pm cpm 8823 Word cword 14468 ⟨“cs1 14549 Basecbs 17148 Σ_g cgsu 17390 Mndcmnd 18659 freeMndcfrmd 18764 mCNcmcn 34749 mVRcmvar 34750 mRExcmrex 34755 mRSubstcmrsub 34759

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13489 df-fzo 13632 df-seq 13971 df-hash 14295 df-word 14469 df-concat 14525 df-s1 14550 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-0g 17391 df-gsum 17392 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-frmd 18766 df-mrex 34775 df-mrsub 34779

This theorem is referenced by: mrsubrn 34802 mrsubff1 34803 mrsub0 34805 mrsubf 34806 mrsubccat 34807 mrsubcn 34808 elmrsubrn 34809 elmsubrn 34817 msubrn 34818 msubff 34819 msubff1 34845

W3C validator