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Theorem msubval 30676
 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 30675 . . 3 ((𝐹:𝐴𝑅𝐴𝑉) → (𝑆𝐹) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩))
763adant3 1074 . 2 ((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) → (𝑆𝐹) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩))
8 simpr 476 . . . 4 (((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) ∧ 𝑒 = 𝑋) → 𝑒 = 𝑋)
98fveq2d 6107 . . 3 (((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) ∧ 𝑒 = 𝑋) → (1st𝑒) = (1st𝑋))
108fveq2d 6107 . . . 4 (((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) ∧ 𝑒 = 𝑋) → (2nd𝑒) = (2nd𝑋))
1110fveq2d 6107 . . 3 (((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) ∧ 𝑒 = 𝑋) → ((𝑂𝐹)‘(2nd𝑒)) = ((𝑂𝐹)‘(2nd𝑋)))
129, 11opeq12d 4348 . 2 (((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) ∧ 𝑒 = 𝑋) → ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩ = ⟨(1st𝑋), ((𝑂𝐹)‘(2nd𝑋))⟩)
13 simp3 1056 . 2 ((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) → 𝑋𝐸)
14 opex 4859 . . 3 ⟨(1st𝑋), ((𝑂𝐹)‘(2nd𝑋))⟩ ∈ V
1514a1i 11 . 2 ((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) → ⟨(1st𝑋), ((𝑂𝐹)‘(2nd𝑋))⟩ ∈ V)
167, 12, 13, 15fvmptd 6197 1 ((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) → ((𝑆𝐹)‘𝑋) = ⟨(1st𝑋), ((𝑂𝐹)‘(2nd𝑋))⟩)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  ⟨cop 4131   ↦ cmpt 4643  ⟶wf 5800  ‘cfv 5804  1st c1st 7057  2nd c2nd 7058  mVRcmvar 30612  mRExcmrex 30617  mExcmex 30618  mRSubstcmrsub 30621  mSubstcmsub 30622 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-pm 7747  df-msub 30642 This theorem is referenced by:  msubrsub  30677  msubty  30678  msubff1  30707
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