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Theorem msubval 34511
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 34510 . . 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 1132 . 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 485 . . . 4 (((𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉 ∧ 𝑋 ∈ 𝐸) ∧ 𝑒 = 𝑋) β†’ 𝑒 = 𝑋)
98fveq2d 6895 . . 3 (((𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉 ∧ 𝑋 ∈ 𝐸) ∧ 𝑒 = 𝑋) β†’ (1st β€˜π‘’) = (1st β€˜π‘‹))
108fveq2d 6895 . . . 4 (((𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉 ∧ 𝑋 ∈ 𝐸) ∧ 𝑒 = 𝑋) β†’ (2nd β€˜π‘’) = (2nd β€˜π‘‹))
1110fveq2d 6895 . . 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 4881 . 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 1138 . 2 ((𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉 ∧ 𝑋 ∈ 𝐸) β†’ 𝑋 ∈ 𝐸)
14 opex 5464 . . 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 7005 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  Vcvv 3474   βŠ† wss 3948  βŸ¨cop 4634   ↦ cmpt 5231  βŸΆwf 6539  β€˜cfv 6543  1st c1st 7972  2nd c2nd 7973  mVRcmvar 34447  mRExcmrex 34452  mExcmex 34453  mRSubstcmrsub 34456  mSubstcmsub 34457
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-pm 8822  df-msub 34477
This theorem is referenced by:  msubrsub  34512  msubty  34513  msubff1  34542
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