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Theorem msubval 35168
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 35167 . . 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 483 . . . 4 (((𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉 ∧ 𝑋 ∈ 𝐸) ∧ 𝑒 = 𝑋) β†’ 𝑒 = 𝑋)
98fveq2d 6906 . . 3 (((𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉 ∧ 𝑋 ∈ 𝐸) ∧ 𝑒 = 𝑋) β†’ (1st β€˜π‘’) = (1st β€˜π‘‹))
108fveq2d 6906 . . . 4 (((𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉 ∧ 𝑋 ∈ 𝐸) ∧ 𝑒 = 𝑋) β†’ (2nd β€˜π‘’) = (2nd β€˜π‘‹))
1110fveq2d 6906 . . 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 4886 . 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 5470 . . 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 7017 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  Vcvv 3473   βŠ† wss 3949  βŸ¨cop 4638   ↦ cmpt 5235  βŸΆwf 6549  β€˜cfv 6553  1st c1st 7997  2nd c2nd 7998  mVRcmvar 35104  mRExcmrex 35109  mExcmex 35110  mRSubstcmrsub 35113  mSubstcmsub 35114
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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-pm 8854  df-msub 35134
This theorem is referenced by:  msubrsub  35169  msubty  35170  msubff1  35199
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