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Theorem msubty 31755
Description: The type of a substituted expression is the same as the original type. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
msubffval.v 𝑉 = (mVR‘𝑇)
msubffval.r 𝑅 = (mREx‘𝑇)
msubffval.s 𝑆 = (mSubst‘𝑇)
msubffval.e 𝐸 = (mEx‘𝑇)
Assertion
Ref Expression
msubty ((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) → (1st ‘((𝑆𝐹)‘𝑋)) = (1st𝑋))

Proof of Theorem msubty
StepHypRef Expression
1 msubffval.v . . 3 𝑉 = (mVR‘𝑇)
2 msubffval.r . . 3 𝑅 = (mREx‘𝑇)
3 msubffval.s . . 3 𝑆 = (mSubst‘𝑇)
4 msubffval.e . . 3 𝐸 = (mEx‘𝑇)
5 eqid 2771 . . 3 (mRSubst‘𝑇) = (mRSubst‘𝑇)
61, 2, 3, 4, 5msubval 31753 . 2 ((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) → ((𝑆𝐹)‘𝑋) = ⟨(1st𝑋), (((mRSubst‘𝑇)‘𝐹)‘(2nd𝑋))⟩)
7 fvex 6340 . . 3 (1st𝑋) ∈ V
8 fvex 6340 . . 3 (((mRSubst‘𝑇)‘𝐹)‘(2nd𝑋)) ∈ V
97, 8op1std 7323 . 2 (((𝑆𝐹)‘𝑋) = ⟨(1st𝑋), (((mRSubst‘𝑇)‘𝐹)‘(2nd𝑋))⟩ → (1st ‘((𝑆𝐹)‘𝑋)) = (1st𝑋))
106, 9syl 17 1 ((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) → (1st ‘((𝑆𝐹)‘𝑋)) = (1st𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1071   = wceq 1631  wcel 2145  wss 3723  cop 4322  wf 6025  cfv 6029  1st c1st 7311  2nd c2nd 7312  mVRcmvar 31689  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
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  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-ral 3066  df-rex 3067  df-reu 3068  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-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  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-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-1st 7313  df-pm 8010  df-msub 31719
This theorem is referenced by: (None)
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