msubff - Mathbox for Mario Carneiro

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Theorem msubff 35498

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 8062	. . . . . . . . 9 ⊢ (𝑒 ∈ ((mTC‘𝑇) × 𝑅) → (1^st ‘𝑒) ∈ (mTC‘𝑇))
2		eqid 2740	. . . . . . . . . 10 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
3		msubff.e	. . . . . . . . . 10 ⊢ 𝐸 = (mEx‘𝑇)
4		msubff.r	. . . . . . . . . 10 ⊢ 𝑅 = (mREx‘𝑇)
5	2, 3, 4	mexval 35470	. . . . . . . . 9 ⊢ 𝐸 = ((mTC‘𝑇) × 𝑅)
6	1, 5	eleq2s 2862	. . . . . . . 8 ⊢ (𝑒 ∈ 𝐸 → (1^st ‘𝑒) ∈ (mTC‘𝑇))
7	6	adantl 481	. . . . . . 7 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) ∧ 𝑒 ∈ 𝐸) → (1^st ‘𝑒) ∈ (mTC‘𝑇))
8		msubff.v	. . . . . . . . . . 11 ⊢ 𝑉 = (mVR‘𝑇)
9		eqid 2740	. . . . . . . . . . 11 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇)
10	8, 4, 9	mrsubff 35480	. . . . . . . . . 10 ⊢ (𝑇 ∈ 𝑊 → (mRSubst‘𝑇):(𝑅 ↑_pm 𝑉)⟶(𝑅 ↑_m 𝑅))
11	10	ffvelcdmda 7118	. . . . . . . . 9 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ (𝑅 ↑_m 𝑅))
12		elmapi 8907	. . . . . . . . 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 8063	. . . . . . . . 9 ⊢ (𝑒 ∈ ((mTC‘𝑇) × 𝑅) → (2^nd ‘𝑒) ∈ 𝑅)
15	14, 5	eleq2s 2862	. . . . . . . 8 ⊢ (𝑒 ∈ 𝐸 → (2^nd ‘𝑒) ∈ 𝑅)
16		ffvelcdm 7115	. . . . . . . 8 ⊢ ((((mRSubst‘𝑇)‘𝑓):𝑅⟶𝑅 ∧ (2^nd ‘𝑒) ∈ 𝑅) → (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒)) ∈ 𝑅)
17	13, 15, 16	syl2an 595	. . . . . . 7 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) ∧ 𝑒 ∈ 𝐸) → (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒)) ∈ 𝑅)
18		opelxp 5736	. . . . . . 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 582	. . . . . 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 2855	. . . . 5 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) ∧ 𝑒 ∈ 𝐸) → ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩ ∈ 𝐸)
21	20	fmpttd 7149	. . . 4 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩):𝐸⟶𝐸)
22	3	fvexi 6934	. . . . 5 ⊢ 𝐸 ∈ V
23	22, 22	elmap 8929	. . . 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 234	. . 3 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑_pm 𝑉)) → (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩) ∈ (𝐸 ↑_m 𝐸))
25	24	fmpttd 7149	. 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 35491	. . 3 ⊢ (𝑇 ∈ 𝑊 → 𝑆 = (𝑓 ∈ (𝑅 ↑_pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘𝑒))⟩)))
28	27	feq1d 6732	. 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 257	1 ⊢ (𝑇 ∈ 𝑊 → 𝑆:(𝑅 ↑_pm 𝑉)⟶(𝐸 ↑_m 𝐸))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ⟨cop 4654 ↦ cmpt 5249 × cxp 5698 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 1^st c1st 8028 2^nd c2nd 8029 ↑_m cmap 8884 ↑_pm cpm 8885 mVRcmvar 35429 mTCcmtc 35432 mRExcmrex 35434 mExcmex 35435 mRSubstcmrsub 35438 mSubstcmsub 35439

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-er 8763 df-map 8886 df-pm 8887 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-n0 12554 df-z 12640 df-uz 12904 df-fz 13568 df-fzo 13712 df-seq 14053 df-hash 14380 df-word 14563 df-concat 14619 df-s1 14644 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-0g 17501 df-gsum 17502 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-frmd 18884 df-mrex 35454 df-mex 35455 df-mrsub 35458 df-msub 35459

This theorem is referenced by: msubf 35500 msubff1 35524 mclsind 35538

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