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Theorem msubty 32995
 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 2759 . . 3 (mRSubst‘𝑇) = (mRSubst‘𝑇)
61, 2, 3, 4, 5msubval 32993 . 2 ((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) → ((𝑆𝐹)‘𝑋) = ⟨(1st𝑋), (((mRSubst‘𝑇)‘𝐹)‘(2nd𝑋))⟩)
7 fvex 6669 . . 3 (1st𝑋) ∈ V
8 fvex 6669 . . 3 (((mRSubst‘𝑇)‘𝐹)‘(2nd𝑋)) ∈ V
97, 8op1std 7701 . 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 1085   = wceq 1539   ∈ wcel 2112   ⊆ wss 3859  ⟨cop 4526  ⟶wf 6329  ‘cfv 6333  1st c1st 7689  2nd c2nd 7690  mVRcmvar 32929  mRExcmrex 32934  mExcmex 32935  mRSubstcmrsub 32938  mSubstcmsub 32939 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7457 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4419  df-pw 4494  df-sn 4521  df-pr 4523  df-op 4527  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5428  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-iota 6292  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7691  df-pm 8417  df-msub 32959 This theorem is referenced by: (None)
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