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Theorem msubff 31135
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 (𝑇𝑊𝑆:(𝑅pm 𝑉)⟶(𝐸𝑚 𝐸))

Proof of Theorem msubff
Dummy variables 𝑒 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xp1st 7143 . . . . . . . . 9 (𝑒 ∈ ((mTC‘𝑇) × 𝑅) → (1st𝑒) ∈ (mTC‘𝑇))
2 eqid 2621 . . . . . . . . . 10 (mTC‘𝑇) = (mTC‘𝑇)
3 msubff.e . . . . . . . . . 10 𝐸 = (mEx‘𝑇)
4 msubff.r . . . . . . . . . 10 𝑅 = (mREx‘𝑇)
52, 3, 4mexval 31107 . . . . . . . . 9 𝐸 = ((mTC‘𝑇) × 𝑅)
61, 5eleq2s 2716 . . . . . . . 8 (𝑒𝐸 → (1st𝑒) ∈ (mTC‘𝑇))
76adantl 482 . . . . . . 7 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝐸) → (1st𝑒) ∈ (mTC‘𝑇))
8 msubff.v . . . . . . . . . . 11 𝑉 = (mVR‘𝑇)
9 eqid 2621 . . . . . . . . . . 11 (mRSubst‘𝑇) = (mRSubst‘𝑇)
108, 4, 9mrsubff 31117 . . . . . . . . . 10 (𝑇𝑊 → (mRSubst‘𝑇):(𝑅pm 𝑉)⟶(𝑅𝑚 𝑅))
1110ffvelrnda 6315 . . . . . . . . 9 ((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ (𝑅𝑚 𝑅))
12 elmapi 7823 . . . . . . . . 9 (((mRSubst‘𝑇)‘𝑓) ∈ (𝑅𝑚 𝑅) → ((mRSubst‘𝑇)‘𝑓):𝑅𝑅)
1311, 12syl 17 . . . . . . . 8 ((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓):𝑅𝑅)
14 xp2nd 7144 . . . . . . . . 9 (𝑒 ∈ ((mTC‘𝑇) × 𝑅) → (2nd𝑒) ∈ 𝑅)
1514, 5eleq2s 2716 . . . . . . . 8 (𝑒𝐸 → (2nd𝑒) ∈ 𝑅)
16 ffvelrn 6313 . . . . . . . 8 ((((mRSubst‘𝑇)‘𝑓):𝑅𝑅 ∧ (2nd𝑒) ∈ 𝑅) → (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒)) ∈ 𝑅)
1713, 15, 16syl2an 494 . . . . . . 7 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝐸) → (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒)) ∈ 𝑅)
18 opelxp 5106 . . . . . . 7 (⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩ ∈ ((mTC‘𝑇) × 𝑅) ↔ ((1st𝑒) ∈ (mTC‘𝑇) ∧ (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒)) ∈ 𝑅))
197, 17, 18sylanbrc 697 . . . . . 6 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝐸) → ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩ ∈ ((mTC‘𝑇) × 𝑅))
2019, 5syl6eleqr 2709 . . . . 5 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝐸) → ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩ ∈ 𝐸)
21 eqid 2621 . . . . 5 (𝑒𝐸 ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩) = (𝑒𝐸 ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)
2220, 21fmptd 6340 . . . 4 ((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) → (𝑒𝐸 ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩):𝐸𝐸)
23 fvex 6158 . . . . . 6 (mEx‘𝑇) ∈ V
243, 23eqeltri 2694 . . . . 5 𝐸 ∈ V
2524, 24elmap 7830 . . . 4 ((𝑒𝐸 ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩) ∈ (𝐸𝑚 𝐸) ↔ (𝑒𝐸 ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩):𝐸𝐸)
2622, 25sylibr 224 . . 3 ((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) → (𝑒𝐸 ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩) ∈ (𝐸𝑚 𝐸))
27 eqid 2621 . . 3 (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)) = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩))
2826, 27fmptd 6340 . 2 (𝑇𝑊 → (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)):(𝑅pm 𝑉)⟶(𝐸𝑚 𝐸))
29 msubff.s . . . 4 𝑆 = (mSubst‘𝑇)
308, 4, 29, 3, 9msubffval 31128 . . 3 (𝑇𝑊𝑆 = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)))
3130feq1d 5987 . 2 (𝑇𝑊 → (𝑆:(𝑅pm 𝑉)⟶(𝐸𝑚 𝐸) ↔ (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)):(𝑅pm 𝑉)⟶(𝐸𝑚 𝐸)))
3228, 31mpbird 247 1 (𝑇𝑊𝑆:(𝑅pm 𝑉)⟶(𝐸𝑚 𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  cop 4154  cmpt 4673   × cxp 5072  wf 5843  cfv 5847  (class class class)co 6604  1st c1st 7111  2nd c2nd 7112  𝑚 cmap 7802  pm cpm 7803  mVRcmvar 31066  mTCcmtc 31069  mRExcmrex 31071  mExcmex 31072  mRSubstcmrsub 31075  mSubstcmsub 31076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407  df-seq 12742  df-hash 13058  df-word 13238  df-concat 13240  df-s1 13241  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-0g 16023  df-gsum 16024  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-submnd 17257  df-frmd 17307  df-mrex 31091  df-mex 31092  df-mrsub 31095  df-msub 31096
This theorem is referenced by:  msubf  31137  msubff1  31161  mclsind  31175
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