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Theorem msubty 31747
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 2806 . . 3 (mRSubst‘𝑇) = (mRSubst‘𝑇)
61, 2, 3, 4, 5msubval 31745 . 2 ((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) → ((𝑆𝐹)‘𝑋) = ⟨(1st𝑋), (((mRSubst‘𝑇)‘𝐹)‘(2nd𝑋))⟩)
7 fvex 6421 . . 3 (1st𝑋) ∈ V
8 fvex 6421 . . 3 (((mRSubst‘𝑇)‘𝐹)‘(2nd𝑋)) ∈ V
97, 8op1std 7408 . 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 1100   = wceq 1637  wcel 2156  wss 3769  cop 4376  wf 6097  cfv 6101  1st c1st 7396  2nd c2nd 7397  mVRcmvar 31681  mRExcmrex 31686  mExcmex 31687  mRSubstcmrsub 31690  mSubstcmsub 31691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-1st 7398  df-pm 8095  df-msub 31711
This theorem is referenced by: (None)
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