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Theorem msubval 35043
Description: A substitution applied to an expression. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
msubffval.v 𝑉 = (mVRβ€˜π‘‡)
msubffval.r 𝑅 = (mRExβ€˜π‘‡)
msubffval.s 𝑆 = (mSubstβ€˜π‘‡)
msubffval.e 𝐸 = (mExβ€˜π‘‡)
msubffval.o 𝑂 = (mRSubstβ€˜π‘‡)
Assertion
Ref Expression
msubval ((𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉 ∧ 𝑋 ∈ 𝐸) β†’ ((π‘†β€˜πΉ)β€˜π‘‹) = ⟨(1st β€˜π‘‹), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘‹))⟩)

Proof of Theorem msubval
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 msubffval.v . . . 4 𝑉 = (mVRβ€˜π‘‡)
2 msubffval.r . . . 4 𝑅 = (mRExβ€˜π‘‡)
3 msubffval.s . . . 4 𝑆 = (mSubstβ€˜π‘‡)
4 msubffval.e . . . 4 𝐸 = (mExβ€˜π‘‡)
5 msubffval.o . . . 4 𝑂 = (mRSubstβ€˜π‘‡)
61, 2, 3, 4, 5msubfval 35042 . . 3 ((𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉) β†’ (π‘†β€˜πΉ) = (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘’))⟩))
763adant3 1129 . 2 ((𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉 ∧ 𝑋 ∈ 𝐸) β†’ (π‘†β€˜πΉ) = (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘’))⟩))
8 simpr 484 . . . 4 (((𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉 ∧ 𝑋 ∈ 𝐸) ∧ 𝑒 = 𝑋) β†’ 𝑒 = 𝑋)
98fveq2d 6888 . . 3 (((𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉 ∧ 𝑋 ∈ 𝐸) ∧ 𝑒 = 𝑋) β†’ (1st β€˜π‘’) = (1st β€˜π‘‹))
108fveq2d 6888 . . . 4 (((𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉 ∧ 𝑋 ∈ 𝐸) ∧ 𝑒 = 𝑋) β†’ (2nd β€˜π‘’) = (2nd β€˜π‘‹))
1110fveq2d 6888 . . 3 (((𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉 ∧ 𝑋 ∈ 𝐸) ∧ 𝑒 = 𝑋) β†’ ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘’)) = ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘‹)))
129, 11opeq12d 4876 . 2 (((𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉 ∧ 𝑋 ∈ 𝐸) ∧ 𝑒 = 𝑋) β†’ ⟨(1st β€˜π‘’), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘’))⟩ = ⟨(1st β€˜π‘‹), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘‹))⟩)
13 simp3 1135 . 2 ((𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉 ∧ 𝑋 ∈ 𝐸) β†’ 𝑋 ∈ 𝐸)
14 opex 5457 . . 3 ⟨(1st β€˜π‘‹), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘‹))⟩ ∈ V
1514a1i 11 . 2 ((𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉 ∧ 𝑋 ∈ 𝐸) β†’ ⟨(1st β€˜π‘‹), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘‹))⟩ ∈ V)
167, 12, 13, 15fvmptd 6998 1 ((𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉 ∧ 𝑋 ∈ 𝐸) β†’ ((π‘†β€˜πΉ)β€˜π‘‹) = ⟨(1st β€˜π‘‹), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘‹))⟩)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  Vcvv 3468   βŠ† wss 3943  βŸ¨cop 4629   ↦ cmpt 5224  βŸΆwf 6532  β€˜cfv 6536  1st c1st 7969  2nd c2nd 7970  mVRcmvar 34979  mRExcmrex 34984  mExcmex 34985  mRSubstcmrsub 34988  mSubstcmsub 34989
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 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
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 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-pm 8822  df-msub 35009
This theorem is referenced by:  msubrsub  35044  msubty  35045  msubff1  35074
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