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Theorem msubval 34971
Description: A substitution applied to an expression. (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)
msubffval.o 𝑂 = (mRSubst‘𝑇)
Assertion
Ref Expression
msubval ((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) → ((𝑆𝐹)‘𝑋) = ⟨(1st𝑋), ((𝑂𝐹)‘(2nd𝑋))⟩)

Proof of Theorem msubval
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 msubffval.v . . . 4 𝑉 = (mVR‘𝑇)
2 msubffval.r . . . 4 𝑅 = (mREx‘𝑇)
3 msubffval.s . . . 4 𝑆 = (mSubst‘𝑇)
4 msubffval.e . . . 4 𝐸 = (mEx‘𝑇)
5 msubffval.o . . . 4 𝑂 = (mRSubst‘𝑇)
61, 2, 3, 4, 5msubfval 34970 . . 3 ((𝐹:𝐴𝑅𝐴𝑉) → (𝑆𝐹) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩))
763adant3 1129 . 2 ((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) → (𝑆𝐹) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩))
8 simpr 484 . . . 4 (((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) ∧ 𝑒 = 𝑋) → 𝑒 = 𝑋)
98fveq2d 6885 . . 3 (((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) ∧ 𝑒 = 𝑋) → (1st𝑒) = (1st𝑋))
108fveq2d 6885 . . . 4 (((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) ∧ 𝑒 = 𝑋) → (2nd𝑒) = (2nd𝑋))
1110fveq2d 6885 . . 3 (((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) ∧ 𝑒 = 𝑋) → ((𝑂𝐹)‘(2nd𝑒)) = ((𝑂𝐹)‘(2nd𝑋)))
129, 11opeq12d 4873 . 2 (((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) ∧ 𝑒 = 𝑋) → ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩ = ⟨(1st𝑋), ((𝑂𝐹)‘(2nd𝑋))⟩)
13 simp3 1135 . 2 ((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) → 𝑋𝐸)
14 opex 5454 . . 3 ⟨(1st𝑋), ((𝑂𝐹)‘(2nd𝑋))⟩ ∈ V
1514a1i 11 . 2 ((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) → ⟨(1st𝑋), ((𝑂𝐹)‘(2nd𝑋))⟩ ∈ V)
167, 12, 13, 15fvmptd 6995 1 ((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) → ((𝑆𝐹)‘𝑋) = ⟨(1st𝑋), ((𝑂𝐹)‘(2nd𝑋))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wcel 2098  Vcvv 3466  wss 3940  cop 4626  cmpt 5221  wf 6529  cfv 6533  1st c1st 7966  2nd c2nd 7967  mVRcmvar 34907  mRExcmrex 34912  mExcmex 34913  mRSubstcmrsub 34916  mSubstcmsub 34917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-pm 8818  df-msub 34937
This theorem is referenced by:  msubrsub  34972  msubty  34973  msubff1  35002
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