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Theorem msubval 33061
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 33060 . . 3 ((𝐹:𝐴𝑅𝐴𝑉) → (𝑆𝐹) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩))
763adant3 1133 . 2 ((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) → (𝑆𝐹) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩))
8 simpr 488 . . . 4 (((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) ∧ 𝑒 = 𝑋) → 𝑒 = 𝑋)
98fveq2d 6681 . . 3 (((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) ∧ 𝑒 = 𝑋) → (1st𝑒) = (1st𝑋))
108fveq2d 6681 . . . 4 (((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) ∧ 𝑒 = 𝑋) → (2nd𝑒) = (2nd𝑋))
1110fveq2d 6681 . . 3 (((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) ∧ 𝑒 = 𝑋) → ((𝑂𝐹)‘(2nd𝑒)) = ((𝑂𝐹)‘(2nd𝑋)))
129, 11opeq12d 4770 . 2 (((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) ∧ 𝑒 = 𝑋) → ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩ = ⟨(1st𝑋), ((𝑂𝐹)‘(2nd𝑋))⟩)
13 simp3 1139 . 2 ((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) → 𝑋𝐸)
14 opex 5323 . . 3 ⟨(1st𝑋), ((𝑂𝐹)‘(2nd𝑋))⟩ ∈ V
1514a1i 11 . 2 ((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) → ⟨(1st𝑋), ((𝑂𝐹)‘(2nd𝑋))⟩ ∈ V)
167, 12, 13, 15fvmptd 6785 1 ((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) → ((𝑆𝐹)‘𝑋) = ⟨(1st𝑋), ((𝑂𝐹)‘(2nd𝑋))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1088   = wceq 1542  wcel 2114  Vcvv 3399  wss 3844  cop 4523  cmpt 5111  wf 6336  cfv 6340  1st c1st 7715  2nd c2nd 7716  mVRcmvar 32997  mRExcmrex 33002  mExcmex 33003  mRSubstcmrsub 33006  mSubstcmsub 33007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7482
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3401  df-sbc 3682  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7176  df-oprab 7177  df-mpo 7178  df-pm 8443  df-msub 33027
This theorem is referenced by:  msubrsub  33062  msubty  33063  msubff1  33092
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