msubff - Mathbox for Mario Carneiro

	Mathbox for Mario Carneiro		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > msubff		Structured version Visualization version GIF version

Theorem msubff 33076

Description: A substitution is a function from 𝐸 to 𝐸. (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)
msubff.e	⊢ 𝐸 = (mEx‘𝑇)

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

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

Step	Hyp	Ref	Expression
1		xp1st 7759	. . . . . . . . 9 ⊢ (𝑒 ∈ ((mTC‘𝑇) × 𝑅) → (1^st ‘𝑒) ∈ (mTC‘𝑇))
2		eqid 2739	. . . . . . . . . 10 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
3		msubff.e	. . . . . . . . . 10 ⊢ 𝐸 = (mEx‘𝑇)
4		msubff.r	. . . . . . . . . 10 ⊢ 𝑅 = (mREx‘𝑇)
5	2, 3, 4	mexval 33048	. . . . . . . . 9 ⊢ 𝐸 = ((mTC‘𝑇) × 𝑅)
6	1, 5	eleq2s 2852	. . . . . . . 8 ⊢ (𝑒 ∈ 𝐸 → (1^st ‘𝑒) ∈ (mTC‘𝑇))
7	6	adantl 485	. . . . . . 7 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) ∧ 𝑒 ∈ 𝐸) → (1^st ‘𝑒) ∈ (mTC‘𝑇))
8		msubff.v	. . . . . . . . . . 11 ⊢ 𝑉 = (mVR‘𝑇)
9		eqid 2739	. . . . . . . . . . 11 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇)
10	8, 4, 9	mrsubff 33058	. . . . . . . . . 10 ⊢ (𝑇 ∈ 𝑊 → (mRSubst‘𝑇):(𝑅 ↑_pm 𝑉)⟶(𝑅 ↑_m 𝑅))
11	10	ffvelrnda 6874	. . . . . . . . 9 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ (𝑅 ↑_m 𝑅))
12		elmapi 8472	. . . . . . . . 9 ⊢ (((mRSubst‘𝑇)‘𝑓) ∈ (𝑅 ↑_m 𝑅) → ((mRSubst‘𝑇)‘𝑓):𝑅⟶𝑅)
13	11, 12	syl 17	. . . . . . . 8 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓):𝑅⟶𝑅)
14		xp2nd 7760	. . . . . . . . 9 ⊢ (𝑒 ∈ ((mTC‘𝑇) × 𝑅) → (2^nd ‘𝑒) ∈ 𝑅)
15	14, 5	eleq2s 2852	. . . . . . . 8 ⊢ (𝑒 ∈ 𝐸 → (2^nd ‘𝑒) ∈ 𝑅)
16		ffvelrn 6872	. . . . . . . 8 ⊢ ((((mRSubst‘𝑇)‘𝑓):𝑅⟶𝑅 ∧ (2^nd ‘𝑒) ∈ 𝑅) → (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒)) ∈ 𝑅)
17	13, 15, 16	syl2an 599	. . . . . . 7 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) ∧ 𝑒 ∈ 𝐸) → (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒)) ∈ 𝑅)
18		opelxp 5571	. . . . . . 7 ⊢ (⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩ ∈ ((mTC‘𝑇) × 𝑅) ↔ ((1^st ‘𝑒) ∈ (mTC‘𝑇) ∧ (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒)) ∈ 𝑅))
19	7, 17, 18	sylanbrc 586	. . . . . 6 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) ∧ 𝑒 ∈ 𝐸) → ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩ ∈ ((mTC‘𝑇) × 𝑅))
20	19, 5	eleqtrrdi 2845	. . . . 5 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) ∧ 𝑒 ∈ 𝐸) → ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩ ∈ 𝐸)
21	20	fmpttd 6902	. . . 4 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩):𝐸⟶𝐸)
22	3	fvexi 6701	. . . . 5 ⊢ 𝐸 ∈ V
23	22, 22	elmap 8494	. . . 4 ⊢ ((𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩) ∈ (𝐸 ↑_m 𝐸) ↔ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩):𝐸⟶𝐸)
24	21, 23	sylibr 237	. . 3 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩) ∈ (𝐸 ↑_m 𝐸))
25	24	fmpttd 6902	. 2 ⊢ (𝑇 ∈ 𝑊 → (𝑓 ∈ (𝑅 ↑_pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩)):(𝑅 ↑_pm 𝑉)⟶(𝐸 ↑_m 𝐸))
26		msubff.s	. . . 4 ⊢ 𝑆 = (mSubst‘𝑇)
27	8, 4, 26, 3, 9	msubffval 33069	. . 3 ⊢ (𝑇 ∈ 𝑊 → 𝑆 = (𝑓 ∈ (𝑅 ↑_pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩)))
28	27	feq1d 6500	. 2 ⊢ (𝑇 ∈ 𝑊 → (𝑆:(𝑅 ↑_pm 𝑉)⟶(𝐸 ↑_m 𝐸) ↔ (𝑓 ∈ (𝑅 ↑_pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩)):(𝑅 ↑_pm 𝑉)⟶(𝐸 ↑_m 𝐸)))
29	25, 28	mpbird 260	1 ⊢ (𝑇 ∈ 𝑊 → 𝑆:(𝑅 ↑_pm 𝑉)⟶(𝐸 ↑_m 𝐸))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ⟨cop 4532 ↦ cmpt 5120 × cxp 5533 ⟶wf 6346 ‘cfv 6350 (class class class)co 7183 1^st c1st 7725 2^nd c2nd 7726 ↑_m cmap 8450 ↑_pm cpm 8451 mVRcmvar 33007 mTCcmtc 33010 mRExcmrex 33012 mExcmex 33013 mRSubstcmrsub 33016 mSubstcmsub 33017

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-cnex 10684 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 ax-pre-mulgt0 10705

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-om 7613 df-1st 7727 df-2nd 7728 df-wrecs 7989 df-recs 8050 df-rdg 8088 df-1o 8144 df-er 8333 df-map 8452 df-pm 8453 df-en 8569 df-dom 8570 df-sdom 8571 df-fin 8572 df-card 9454 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-sub 10963 df-neg 10964 df-nn 11730 df-2 11792 df-n0 11990 df-z 12076 df-uz 12338 df-fz 12995 df-fzo 13138 df-seq 13474 df-hash 13796 df-word 13969 df-concat 14025 df-s1 14052 df-struct 16601 df-ndx 16602 df-slot 16603 df-base 16605 df-sets 16606 df-ress 16607 df-plusg 16694 df-0g 16831 df-gsum 16832 df-mgm 17981 df-sgrp 18030 df-mnd 18041 df-submnd 18086 df-frmd 18143 df-mrex 33032 df-mex 33033 df-mrsub 33036 df-msub 33037

This theorem is referenced by: msubf 33078 msubff1 33102 mclsind 33116

W3C validator