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Theorem msubval 34119
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 34118 . . 3 ((𝐹:𝐴𝑅𝐴𝑉) → (𝑆𝐹) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩))
763adant3 1132 . 2 ((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) → (𝑆𝐹) = (𝑒𝐸 ↦ ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩))
8 simpr 485 . . . 4 (((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) ∧ 𝑒 = 𝑋) → 𝑒 = 𝑋)
98fveq2d 6846 . . 3 (((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) ∧ 𝑒 = 𝑋) → (1st𝑒) = (1st𝑋))
108fveq2d 6846 . . . 4 (((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) ∧ 𝑒 = 𝑋) → (2nd𝑒) = (2nd𝑋))
1110fveq2d 6846 . . 3 (((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) ∧ 𝑒 = 𝑋) → ((𝑂𝐹)‘(2nd𝑒)) = ((𝑂𝐹)‘(2nd𝑋)))
129, 11opeq12d 4838 . 2 (((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) ∧ 𝑒 = 𝑋) → ⟨(1st𝑒), ((𝑂𝐹)‘(2nd𝑒))⟩ = ⟨(1st𝑋), ((𝑂𝐹)‘(2nd𝑋))⟩)
13 simp3 1138 . 2 ((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) → 𝑋𝐸)
14 opex 5421 . . 3 ⟨(1st𝑋), ((𝑂𝐹)‘(2nd𝑋))⟩ ∈ V
1514a1i 11 . 2 ((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) → ⟨(1st𝑋), ((𝑂𝐹)‘(2nd𝑋))⟩ ∈ V)
167, 12, 13, 15fvmptd 6955 1 ((𝐹:𝐴𝑅𝐴𝑉𝑋𝐸) → ((𝑆𝐹)‘𝑋) = ⟨(1st𝑋), ((𝑂𝐹)‘(2nd𝑋))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3445  wss 3910  cop 4592  cmpt 5188  wf 6492  cfv 6496  1st c1st 7919  2nd c2nd 7920  mVRcmvar 34055  mRExcmrex 34060  mExcmex 34061  mRSubstcmrsub 34064  mSubstcmsub 34065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-pm 8768  df-msub 34085
This theorem is referenced by:  msubrsub  34120  msubty  34121  msubff1  34150
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