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Theorem msubval 31760
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 31759 . . 3 ((𝐹:𝐴𝑅𝐴𝑉) → (𝑆𝐹) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩))
763adant3 1126 . 2 ((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) → (𝑆𝐹) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩))
8 simpr 471 . . . 4 (((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) ∧ 𝑒 = 𝑋) → 𝑒 = 𝑋)
98fveq2d 6336 . . 3 (((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) ∧ 𝑒 = 𝑋) → (1st𝑒) = (1st𝑋))
108fveq2d 6336 . . . 4 (((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) ∧ 𝑒 = 𝑋) → (2nd𝑒) = (2nd𝑋))
1110fveq2d 6336 . . 3 (((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) ∧ 𝑒 = 𝑋) → ((𝑂𝐹)‘(2nd𝑒)) = ((𝑂𝐹)‘(2nd𝑋)))
129, 11opeq12d 4547 . 2 (((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) ∧ 𝑒 = 𝑋) → ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩ = ⟨(1st𝑋), ((𝑂𝐹)‘(2nd𝑋))⟩)
13 simp3 1132 . 2 ((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) → 𝑋𝐸)
14 opex 5060 . . 3 ⟨(1st𝑋), ((𝑂𝐹)‘(2nd𝑋))⟩ ∈ V
1514a1i 11 . 2 ((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) → ⟨(1st𝑋), ((𝑂𝐹)‘(2nd𝑋))⟩ ∈ V)
167, 12, 13, 15fvmptd 6430 1 ((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) → ((𝑆𝐹)‘𝑋) = ⟨(1st𝑋), ((𝑂𝐹)‘(2nd𝑋))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  Vcvv 3351  wss 3723  cop 4322  cmpt 4863  wf 6027  cfv 6031  1st c1st 7313  2nd c2nd 7314  mVRcmvar 31696  mRExcmrex 31701  mExcmex 31702  mRSubstcmrsub 31705  mSubstcmsub 31706
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 7096
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 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-pm 8012  df-msub 31726
This theorem is referenced by:  msubrsub  31761  msubty  31762  msubff1  31791
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