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Theorem msubfval 34182
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
msubfval ((𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉) β†’ (π‘†β€˜πΉ) = (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘’))⟩))
Distinct variable groups:   𝑒,𝐸   𝑒,𝑂   𝑅,𝑒   𝑇,𝑒   𝑒,𝑉   𝐴,𝑒   𝑒,𝐹
Allowed substitution hint:   𝑆(𝑒)

Proof of Theorem msubfval
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 msubffval.v . . . . . 6 𝑉 = (mVRβ€˜π‘‡)
2 msubffval.r . . . . . 6 𝑅 = (mRExβ€˜π‘‡)
3 msubffval.s . . . . . 6 𝑆 = (mSubstβ€˜π‘‡)
4 msubffval.e . . . . . 6 𝐸 = (mExβ€˜π‘‡)
5 msubffval.o . . . . . 6 𝑂 = (mRSubstβ€˜π‘‡)
61, 2, 3, 4, 5msubffval 34181 . . . . 5 (𝑇 ∈ V β†’ 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜π‘“)β€˜(2nd β€˜π‘’))⟩)))
76adantr 482 . . . 4 ((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) β†’ 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜π‘“)β€˜(2nd β€˜π‘’))⟩)))
8 simplr 768 . . . . . . . 8 ((((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒 ∈ 𝐸) β†’ 𝑓 = 𝐹)
98fveq2d 6850 . . . . . . 7 ((((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒 ∈ 𝐸) β†’ (π‘‚β€˜π‘“) = (π‘‚β€˜πΉ))
109fveq1d 6848 . . . . . 6 ((((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒 ∈ 𝐸) β†’ ((π‘‚β€˜π‘“)β€˜(2nd β€˜π‘’)) = ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘’)))
1110opeq2d 4841 . . . . 5 ((((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒 ∈ 𝐸) β†’ ⟨(1st β€˜π‘’), ((π‘‚β€˜π‘“)β€˜(2nd β€˜π‘’))⟩ = ⟨(1st β€˜π‘’), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘’))⟩)
1211mpteq2dva 5209 . . . 4 (((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) ∧ 𝑓 = 𝐹) β†’ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜π‘“)β€˜(2nd β€˜π‘’))⟩) = (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘’))⟩))
132fvexi 6860 . . . . . . 7 𝑅 ∈ V
141fvexi 6860 . . . . . . 7 𝑉 ∈ V
1513, 14pm3.2i 472 . . . . . 6 (𝑅 ∈ V ∧ 𝑉 ∈ V)
1615a1i 11 . . . . 5 (𝑇 ∈ V β†’ (𝑅 ∈ V ∧ 𝑉 ∈ V))
17 elpm2r 8789 . . . . 5 (((𝑅 ∈ V ∧ 𝑉 ∈ V) ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) β†’ 𝐹 ∈ (𝑅 ↑pm 𝑉))
1816, 17sylan 581 . . . 4 ((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) β†’ 𝐹 ∈ (𝑅 ↑pm 𝑉))
194fvexi 6860 . . . . . 6 𝐸 ∈ V
2019mptex 7177 . . . . 5 (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘’))⟩) ∈ V
2120a1i 11 . . . 4 ((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) β†’ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘’))⟩) ∈ V)
227, 12, 18, 21fvmptd 6959 . . 3 ((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) β†’ (π‘†β€˜πΉ) = (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘’))⟩))
2322ex 414 . 2 (𝑇 ∈ V β†’ ((𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉) β†’ (π‘†β€˜πΉ) = (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘’))⟩)))
24 0fv 6890 . . . . 5 (βˆ…β€˜πΉ) = βˆ…
25 mpt0 6647 . . . . 5 (𝑒 ∈ βˆ… ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘’))⟩) = βˆ…
2624, 25eqtr4i 2764 . . . 4 (βˆ…β€˜πΉ) = (𝑒 ∈ βˆ… ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘’))⟩)
27 fvprc 6838 . . . . . 6 (Β¬ 𝑇 ∈ V β†’ (mSubstβ€˜π‘‡) = βˆ…)
283, 27eqtrid 2785 . . . . 5 (Β¬ 𝑇 ∈ V β†’ 𝑆 = βˆ…)
2928fveq1d 6848 . . . 4 (Β¬ 𝑇 ∈ V β†’ (π‘†β€˜πΉ) = (βˆ…β€˜πΉ))
30 fvprc 6838 . . . . . 6 (Β¬ 𝑇 ∈ V β†’ (mExβ€˜π‘‡) = βˆ…)
314, 30eqtrid 2785 . . . . 5 (Β¬ 𝑇 ∈ V β†’ 𝐸 = βˆ…)
3231mpteq1d 5204 . . . 4 (Β¬ 𝑇 ∈ V β†’ (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘’))⟩) = (𝑒 ∈ βˆ… ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘’))⟩))
3326, 29, 323eqtr4a 2799 . . 3 (Β¬ 𝑇 ∈ V β†’ (π‘†β€˜πΉ) = (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘’))⟩))
3433a1d 25 . 2 (Β¬ 𝑇 ∈ V β†’ ((𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉) β†’ (π‘†β€˜πΉ) = (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘’))⟩)))
3523, 34pm2.61i 182 1 ((𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉) β†’ (π‘†β€˜πΉ) = (𝑒 ∈ 𝐸 ↦ ⟨(1st β€˜π‘’), ((π‘‚β€˜πΉ)β€˜(2nd β€˜π‘’))⟩))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3447   βŠ† wss 3914  βˆ…c0 4286  βŸ¨cop 4596   ↦ cmpt 5192  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361  1st c1st 7923  2nd c2nd 7924   ↑pm cpm 8772  mVRcmvar 34119  mRExcmrex 34124  mExcmex 34125  mRSubstcmrsub 34128  mSubstcmsub 34129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-pm 8774  df-msub 34149
This theorem is referenced by:  msubval  34183  msubrn  34187
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