Theorem msubff 36035

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 8020	. . . . . . . . 9 ⊢ (𝑒 ∈ ((mTC‘𝑇) × 𝑅) → (1st ‘𝑒) ∈ (mTC‘𝑇))
2		eqid 2765	. . . . . . . . . 10 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
3		msubff.e	. . . . . . . . . 10 ⊢ 𝐸 = (mEx‘𝑇)
4		msubff.r	. . . . . . . . . 10 ⊢ 𝑅 = (mREx‘𝑇)
5	2, 3, 4	mexval 36007	. . . . . . . . 9 ⊢ 𝐸 = ((mTC‘𝑇) × 𝑅)
6	1, 5	eleq2s 2883	. . . . . . . 8 ⊢ (𝑒 ∈ 𝐸 → (1st ‘𝑒) ∈ (mTC‘𝑇))
7	6	adantl 487	. . . . . . 7 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑒 ∈ 𝐸) → (1st ‘𝑒) ∈ (mTC‘𝑇))
8		msubff.v	. . . . . . . . . . 11 ⊢ 𝑉 = (mVR‘𝑇)
9		eqid 2765	. . . . . . . . . . 11 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇)
10	8, 4, 9	mrsubff 36017	. . . . . . . . . 10 ⊢ (𝑇 ∈ 𝑊 → (mRSubst‘𝑇):(𝑅 ↑pm 𝑉)⟶(𝑅 ↑m 𝑅))
11	10	ffvelcdmda 7083	. . . . . . . . 9 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ (𝑅 ↑m 𝑅))
12		elmapi 8848	. . . . . . . . 9 ⊢ (((mRSubst‘𝑇)‘𝑓) ∈ (𝑅 ↑m 𝑅) → ((mRSubst‘𝑇)‘𝑓):𝑅⟶𝑅)
13	11, 12	syl 18	. . . . . . . 8 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓):𝑅⟶𝑅)
14		xp2nd 8021	. . . . . . . . 9 ⊢ (𝑒 ∈ ((mTC‘𝑇) × 𝑅) → (2nd ‘𝑒) ∈ 𝑅)
15	14, 5	eleq2s 2883	. . . . . . . 8 ⊢ (𝑒 ∈ 𝐸 → (2nd ‘𝑒) ∈ 𝑅)
16		ffvelcdm 7080	. . . . . . . 8 ⊢ ((((mRSubst‘𝑇)‘𝑓):𝑅⟶𝑅 ∧ (2nd ‘𝑒) ∈ 𝑅) → (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒)) ∈ 𝑅)
17	13, 15, 16	syl2an 608	. . . . . . 7 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑒 ∈ 𝐸) → (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒)) ∈ 𝑅)
18		opelxp 5699	. . . . . . 7 ⊢ (⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩ ∈ ((mTC‘𝑇) × 𝑅) ↔ ((1st ‘𝑒) ∈ (mTC‘𝑇) ∧ (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒)) ∈ 𝑅))
19	7, 17, 18	sylanbrc 595	. . . . . 6 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑒 ∈ 𝐸) → ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩ ∈ ((mTC‘𝑇) × 𝑅))
20	19, 5	eleqtrrdi 2876	. . . . 5 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑒 ∈ 𝐸) → ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩ ∈ 𝐸)
21	20	fmpttd 7114	. . . 4 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩):𝐸⟶𝐸)
22	3	fvexi 6899	. . . . 5 ⊢ 𝐸 ∈ V
23	22, 22	elmap 8871	. . . 4 ⊢ ((𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩) ∈ (𝐸 ↑m 𝐸) ↔ (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩):𝐸⟶𝐸)
24	21, 23	sylibr 237	. . 3 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩) ∈ (𝐸 ↑m 𝐸))
25	24	fmpttd 7114	. 2 ⊢ (𝑇 ∈ 𝑊 → (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩)):(𝑅 ↑pm 𝑉)⟶(𝐸 ↑m 𝐸))
26		msubff.s	. . . 4 ⊢ 𝑆 = (mSubst‘𝑇)
27	8, 4, 26, 3, 9	msubffval 36028	. . 3 ⊢ (𝑇 ∈ 𝑊 → 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩)))
28	27	feq1d 6691	. 2 ⊢ (𝑇 ∈ 𝑊 → (𝑆:(𝑅 ↑pm 𝑉)⟶(𝐸 ↑m 𝐸) ↔ (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))⟩)):(𝑅 ↑pm 𝑉)⟶(𝐸 ↑m 𝐸)))
29	25, 28	mpbird 260	1 ⊢ (𝑇 ∈ 𝑊 → 𝑆:(𝑅 ↑pm 𝑉)⟶(𝐸 ↑m 𝐸))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ⟨cop 4597   ↦ cmpt 5194   × cxp 5661  ⟶wf 6536  ‘cfv 6540  (class class class)co 7416  1^st c1st 7986  2^nd c2nd 7987   ↑_m cmap 8826   ↑_pm cpm 8827  mVRcmvar 35966  mTCcmtc 35969  mRExcmrex 35971  mExcmex 35972  mRSubstcmrsub 35975  mSubstcmsub 35976

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738  ax-cnex 11167  ax-resscn 11168  ax-1cn 11169  ax-icn 11170  ax-addcl 11171  ax-addrcl 11172  ax-mulcl 11173  ax-mulrcl 11174  ax-mulcom 11175  ax-addass 11176  ax-mulass 11177  ax-distr 11178  ax-i2m1 11179  ax-1ne0 11180  ax-1rid 11181  ax-rnegex 11182  ax-rrecex 11183  ax-cnre 11184  ax-pre-lttri 11185  ax-pre-lttrn 11186  ax-pre-ltadd 11187  ax-pre-mulgt0 11188

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-er 8696  df-map 8828  df-pm 8829  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-card 9937  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11454  df-neg 11455  df-nn 12245  df-2 12314  df-n0 12516  df-z 12603  df-uz 12875  df-fz 13548  df-fzo 13696  df-seq 14052  df-hash 14381  df-word 14565  df-concat 14622  df-s1 14649  df-struct 17225  df-sets 17242  df-slot 17260  df-ndx 17272  df-base 17288  df-ress 17309  df-plusg 17341  df-0g 17512  df-gsum 17513  df-mgm 18716  df-sgrp 18799  df-mnd 18815  df-submnd 18866  df-frmd 18932  df-mrex 35991  df-mex 35992  df-mrsub 35995  df-msub 35996

This theorem is used by:  msubf  36037  msubff1  36061  mclsind  36075