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Theorem msubff 31758
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 7345 . . . . . . . . 9 (𝑒 ∈ ((mTC‘𝑇) × 𝑅) → (1st𝑒) ∈ (mTC‘𝑇))
2 eqid 2771 . . . . . . . . . 10 (mTC‘𝑇) = (mTC‘𝑇)
3 msubff.e . . . . . . . . . 10 𝐸 = (mEx‘𝑇)
4 msubff.r . . . . . . . . . 10 𝑅 = (mREx‘𝑇)
52, 3, 4mexval 31730 . . . . . . . . 9 𝐸 = ((mTC‘𝑇) × 𝑅)
61, 5eleq2s 2868 . . . . . . . 8 (𝑒𝐸 → (1st𝑒) ∈ (mTC‘𝑇))
76adantl 467 . . . . . . 7 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝐸) → (1st𝑒) ∈ (mTC‘𝑇))
8 msubff.v . . . . . . . . . . 11 𝑉 = (mVR‘𝑇)
9 eqid 2771 . . . . . . . . . . 11 (mRSubst‘𝑇) = (mRSubst‘𝑇)
108, 4, 9mrsubff 31740 . . . . . . . . . 10 (𝑇𝑊 → (mRSubst‘𝑇):(𝑅pm 𝑉)⟶(𝑅𝑚 𝑅))
1110ffvelrnda 6500 . . . . . . . . 9 ((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ (𝑅𝑚 𝑅))
12 elmapi 8029 . . . . . . . . 9 (((mRSubst‘𝑇)‘𝑓) ∈ (𝑅𝑚 𝑅) → ((mRSubst‘𝑇)‘𝑓):𝑅𝑅)
1311, 12syl 17 . . . . . . . 8 ((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓):𝑅𝑅)
14 xp2nd 7346 . . . . . . . . 9 (𝑒 ∈ ((mTC‘𝑇) × 𝑅) → (2nd𝑒) ∈ 𝑅)
1514, 5eleq2s 2868 . . . . . . . 8 (𝑒𝐸 → (2nd𝑒) ∈ 𝑅)
16 ffvelrn 6498 . . . . . . . 8 ((((mRSubst‘𝑇)‘𝑓):𝑅𝑅 ∧ (2nd𝑒) ∈ 𝑅) → (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒)) ∈ 𝑅)
1713, 15, 16syl2an 583 . . . . . . 7 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝐸) → (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒)) ∈ 𝑅)
18 opelxp 5284 . . . . . . 7 (⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩ ∈ ((mTC‘𝑇) × 𝑅) ↔ ((1st𝑒) ∈ (mTC‘𝑇) ∧ (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒)) ∈ 𝑅))
197, 17, 18sylanbrc 572 . . . . . 6 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝐸) → ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩ ∈ ((mTC‘𝑇) × 𝑅))
2019, 5syl6eleqr 2861 . . . . 5 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝐸) → ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩ ∈ 𝐸)
21 eqid 2771 . . . . 5 (𝑒𝐸 ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩) = (𝑒𝐸 ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)
2220, 21fmptd 6525 . . . 4 ((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) → (𝑒𝐸 ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩):𝐸𝐸)
233fvexi 6341 . . . . 5 𝐸 ∈ V
2423, 23elmap 8036 . . . 4 ((𝑒𝐸 ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩) ∈ (𝐸𝑚 𝐸) ↔ (𝑒𝐸 ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩):𝐸𝐸)
2522, 24sylibr 224 . . 3 ((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) → (𝑒𝐸 ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩) ∈ (𝐸𝑚 𝐸))
26 eqid 2771 . . 3 (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)) = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩))
2725, 26fmptd 6525 . 2 (𝑇𝑊 → (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)):(𝑅pm 𝑉)⟶(𝐸𝑚 𝐸))
28 msubff.s . . . 4 𝑆 = (mSubst‘𝑇)
298, 4, 28, 3, 9msubffval 31751 . . 3 (𝑇𝑊𝑆 = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)))
3029feq1d 6168 . 2 (𝑇𝑊 → (𝑆:(𝑅pm 𝑉)⟶(𝐸𝑚 𝐸) ↔ (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝐸 ↦ ⟨(1st𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd𝑒))⟩)):(𝑅pm 𝑉)⟶(𝐸𝑚 𝐸)))
3127, 30mpbird 247 1 (𝑇𝑊𝑆:(𝑅pm 𝑉)⟶(𝐸𝑚 𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  cop 4322  cmpt 4863   × cxp 5247  wf 6025  cfv 6029  (class class class)co 6791  1st c1st 7311  2nd c2nd 7312  𝑚 cmap 8007  pm cpm 8008  mVRcmvar 31689  mTCcmtc 31692  mRExcmrex 31694  mExcmex 31695  mRSubstcmrsub 31698  mSubstcmsub 31699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094  ax-cnex 10192  ax-resscn 10193  ax-1cn 10194  ax-icn 10195  ax-addcl 10196  ax-addrcl 10197  ax-mulcl 10198  ax-mulrcl 10199  ax-mulcom 10200  ax-addass 10201  ax-mulass 10202  ax-distr 10203  ax-i2m1 10204  ax-1ne0 10205  ax-1rid 10206  ax-rnegex 10207  ax-rrecex 10208  ax-cnre 10209  ax-pre-lttri 10210  ax-pre-lttrn 10211  ax-pre-ltadd 10212  ax-pre-mulgt0 10213
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5821  df-ord 5867  df-on 5868  df-lim 5869  df-suc 5870  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-riota 6752  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-om 7211  df-1st 7313  df-2nd 7314  df-wrecs 7557  df-recs 7619  df-rdg 7657  df-1o 7711  df-oadd 7715  df-er 7894  df-map 8009  df-pm 8010  df-en 8108  df-dom 8109  df-sdom 8110  df-fin 8111  df-card 8963  df-pnf 10276  df-mnf 10277  df-xr 10278  df-ltxr 10279  df-le 10280  df-sub 10468  df-neg 10469  df-nn 11221  df-2 11279  df-n0 11493  df-z 11578  df-uz 11887  df-fz 12527  df-fzo 12667  df-seq 13002  df-hash 13315  df-word 13488  df-concat 13490  df-s1 13491  df-struct 16059  df-ndx 16060  df-slot 16061  df-base 16063  df-sets 16064  df-ress 16065  df-plusg 16155  df-0g 16303  df-gsum 16304  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-submnd 17537  df-frmd 17587  df-mrex 31714  df-mex 31715  df-mrsub 31718  df-msub 31719
This theorem is referenced by:  msubf  31760  msubff1  31784  mclsind  31798
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