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Theorem mrsubfval 34499
Description: The substitution of some variables for expressions in a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mrsubffval.c 𝐢 = (mCNβ€˜π‘‡)
mrsubffval.v 𝑉 = (mVRβ€˜π‘‡)
mrsubffval.r 𝑅 = (mRExβ€˜π‘‡)
mrsubffval.s 𝑆 = (mRSubstβ€˜π‘‡)
mrsubffval.g 𝐺 = (freeMndβ€˜(𝐢 βˆͺ 𝑉))
Assertion
Ref Expression
mrsubfval ((𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉) β†’ (π‘†β€˜πΉ) = (𝑒 ∈ 𝑅 ↦ (𝐺 Ξ£g ((𝑣 ∈ (𝐢 βˆͺ 𝑉) ↦ if(𝑣 ∈ 𝐴, (πΉβ€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒))))
Distinct variable groups:   𝑣,𝑒,𝐴   𝐢,𝑒,𝑣   𝑒,𝐹,𝑣   𝑅,𝑒,𝑣   𝑒,𝐺   𝑇,𝑒,𝑣   𝑒,𝑉,𝑣
Allowed substitution hints:   𝑆(𝑣,𝑒)   𝐺(𝑣)

Proof of Theorem mrsubfval
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 mrsubffval.c . . . . . 6 𝐢 = (mCNβ€˜π‘‡)
2 mrsubffval.v . . . . . 6 𝑉 = (mVRβ€˜π‘‡)
3 mrsubffval.r . . . . . 6 𝑅 = (mRExβ€˜π‘‡)
4 mrsubffval.s . . . . . 6 𝑆 = (mRSubstβ€˜π‘‡)
5 mrsubffval.g . . . . . 6 𝐺 = (freeMndβ€˜(𝐢 βˆͺ 𝑉))
61, 2, 3, 4, 5mrsubffval 34498 . . . . 5 (𝑇 ∈ V β†’ 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝑅 ↦ (𝐺 Ξ£g ((𝑣 ∈ (𝐢 βˆͺ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (π‘“β€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒)))))
76adantr 482 . . . 4 ((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) β†’ 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝑅 ↦ (𝐺 Ξ£g ((𝑣 ∈ (𝐢 βˆͺ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (π‘“β€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒)))))
8 dmeq 5904 . . . . . . . . . . 11 (𝑓 = 𝐹 β†’ dom 𝑓 = dom 𝐹)
9 fdm 6727 . . . . . . . . . . . 12 (𝐹:π΄βŸΆπ‘… β†’ dom 𝐹 = 𝐴)
109ad2antrl 727 . . . . . . . . . . 11 ((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) β†’ dom 𝐹 = 𝐴)
118, 10sylan9eqr 2795 . . . . . . . . . 10 (((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) ∧ 𝑓 = 𝐹) β†’ dom 𝑓 = 𝐴)
1211eleq2d 2820 . . . . . . . . 9 (((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) ∧ 𝑓 = 𝐹) β†’ (𝑣 ∈ dom 𝑓 ↔ 𝑣 ∈ 𝐴))
13 simpr 486 . . . . . . . . . 10 (((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) ∧ 𝑓 = 𝐹) β†’ 𝑓 = 𝐹)
1413fveq1d 6894 . . . . . . . . 9 (((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) ∧ 𝑓 = 𝐹) β†’ (π‘“β€˜π‘£) = (πΉβ€˜π‘£))
1512, 14ifbieq1d 4553 . . . . . . . 8 (((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) ∧ 𝑓 = 𝐹) β†’ if(𝑣 ∈ dom 𝑓, (π‘“β€˜π‘£), βŸ¨β€œπ‘£β€βŸ©) = if(𝑣 ∈ 𝐴, (πΉβ€˜π‘£), βŸ¨β€œπ‘£β€βŸ©))
1615mpteq2dv 5251 . . . . . . 7 (((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) ∧ 𝑓 = 𝐹) β†’ (𝑣 ∈ (𝐢 βˆͺ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (π‘“β€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) = (𝑣 ∈ (𝐢 βˆͺ 𝑉) ↦ if(𝑣 ∈ 𝐴, (πΉβ€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)))
1716coeq1d 5862 . . . . . 6 (((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) ∧ 𝑓 = 𝐹) β†’ ((𝑣 ∈ (𝐢 βˆͺ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (π‘“β€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒) = ((𝑣 ∈ (𝐢 βˆͺ 𝑉) ↦ if(𝑣 ∈ 𝐴, (πΉβ€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒))
1817oveq2d 7425 . . . . 5 (((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) ∧ 𝑓 = 𝐹) β†’ (𝐺 Ξ£g ((𝑣 ∈ (𝐢 βˆͺ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (π‘“β€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒)) = (𝐺 Ξ£g ((𝑣 ∈ (𝐢 βˆͺ 𝑉) ↦ if(𝑣 ∈ 𝐴, (πΉβ€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒)))
1918mpteq2dv 5251 . . . 4 (((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) ∧ 𝑓 = 𝐹) β†’ (𝑒 ∈ 𝑅 ↦ (𝐺 Ξ£g ((𝑣 ∈ (𝐢 βˆͺ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (π‘“β€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒))) = (𝑒 ∈ 𝑅 ↦ (𝐺 Ξ£g ((𝑣 ∈ (𝐢 βˆͺ 𝑉) ↦ if(𝑣 ∈ 𝐴, (πΉβ€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒))))
203fvexi 6906 . . . . . 6 𝑅 ∈ V
2120a1i 11 . . . . 5 ((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) β†’ 𝑅 ∈ V)
222fvexi 6906 . . . . . 6 𝑉 ∈ V
2322a1i 11 . . . . 5 ((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) β†’ 𝑉 ∈ V)
24 simprl 770 . . . . 5 ((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) β†’ 𝐹:π΄βŸΆπ‘…)
25 simprr 772 . . . . 5 ((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) β†’ 𝐴 βŠ† 𝑉)
26 elpm2r 8839 . . . . 5 (((𝑅 ∈ V ∧ 𝑉 ∈ V) ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) β†’ 𝐹 ∈ (𝑅 ↑pm 𝑉))
2721, 23, 24, 25, 26syl22anc 838 . . . 4 ((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) β†’ 𝐹 ∈ (𝑅 ↑pm 𝑉))
2820mptex 7225 . . . . 5 (𝑒 ∈ 𝑅 ↦ (𝐺 Ξ£g ((𝑣 ∈ (𝐢 βˆͺ 𝑉) ↦ if(𝑣 ∈ 𝐴, (πΉβ€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒))) ∈ V
2928a1i 11 . . . 4 ((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) β†’ (𝑒 ∈ 𝑅 ↦ (𝐺 Ξ£g ((𝑣 ∈ (𝐢 βˆͺ 𝑉) ↦ if(𝑣 ∈ 𝐴, (πΉβ€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒))) ∈ V)
307, 19, 27, 29fvmptd 7006 . . 3 ((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) β†’ (π‘†β€˜πΉ) = (𝑒 ∈ 𝑅 ↦ (𝐺 Ξ£g ((𝑣 ∈ (𝐢 βˆͺ 𝑉) ↦ if(𝑣 ∈ 𝐴, (πΉβ€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒))))
3130ex 414 . 2 (𝑇 ∈ V β†’ ((𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉) β†’ (π‘†β€˜πΉ) = (𝑒 ∈ 𝑅 ↦ (𝐺 Ξ£g ((𝑣 ∈ (𝐢 βˆͺ 𝑉) ↦ if(𝑣 ∈ 𝐴, (πΉβ€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒)))))
32 0fv 6936 . . . 4 (βˆ…β€˜πΉ) = βˆ…
33 fvprc 6884 . . . . . 6 (Β¬ 𝑇 ∈ V β†’ (mRSubstβ€˜π‘‡) = βˆ…)
344, 33eqtrid 2785 . . . . 5 (Β¬ 𝑇 ∈ V β†’ 𝑆 = βˆ…)
3534fveq1d 6894 . . . 4 (Β¬ 𝑇 ∈ V β†’ (π‘†β€˜πΉ) = (βˆ…β€˜πΉ))
36 fvprc 6884 . . . . . . 7 (Β¬ 𝑇 ∈ V β†’ (mRExβ€˜π‘‡) = βˆ…)
373, 36eqtrid 2785 . . . . . 6 (Β¬ 𝑇 ∈ V β†’ 𝑅 = βˆ…)
3837mpteq1d 5244 . . . . 5 (Β¬ 𝑇 ∈ V β†’ (𝑒 ∈ 𝑅 ↦ (𝐺 Ξ£g ((𝑣 ∈ (𝐢 βˆͺ 𝑉) ↦ if(𝑣 ∈ 𝐴, (πΉβ€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒))) = (𝑒 ∈ βˆ… ↦ (𝐺 Ξ£g ((𝑣 ∈ (𝐢 βˆͺ 𝑉) ↦ if(𝑣 ∈ 𝐴, (πΉβ€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒))))
39 mpt0 6693 . . . . 5 (𝑒 ∈ βˆ… ↦ (𝐺 Ξ£g ((𝑣 ∈ (𝐢 βˆͺ 𝑉) ↦ if(𝑣 ∈ 𝐴, (πΉβ€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒))) = βˆ…
4038, 39eqtrdi 2789 . . . 4 (Β¬ 𝑇 ∈ V β†’ (𝑒 ∈ 𝑅 ↦ (𝐺 Ξ£g ((𝑣 ∈ (𝐢 βˆͺ 𝑉) ↦ if(𝑣 ∈ 𝐴, (πΉβ€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒))) = βˆ…)
4132, 35, 403eqtr4a 2799 . . 3 (Β¬ 𝑇 ∈ V β†’ (π‘†β€˜πΉ) = (𝑒 ∈ 𝑅 ↦ (𝐺 Ξ£g ((𝑣 ∈ (𝐢 βˆͺ 𝑉) ↦ if(𝑣 ∈ 𝐴, (πΉβ€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒))))
4241a1d 25 . 2 (Β¬ 𝑇 ∈ V β†’ ((𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉) β†’ (π‘†β€˜πΉ) = (𝑒 ∈ 𝑅 ↦ (𝐺 Ξ£g ((𝑣 ∈ (𝐢 βˆͺ 𝑉) ↦ if(𝑣 ∈ 𝐴, (πΉβ€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒)))))
4331, 42pm2.61i 182 1 ((𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉) β†’ (π‘†β€˜πΉ) = (𝑒 ∈ 𝑅 ↦ (𝐺 Ξ£g ((𝑣 ∈ (𝐢 βˆͺ 𝑉) ↦ if(𝑣 ∈ 𝐴, (πΉβ€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3475   βˆͺ cun 3947   βŠ† wss 3949  βˆ…c0 4323  ifcif 4529   ↦ cmpt 5232  dom cdm 5677   ∘ ccom 5681  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ↑pm cpm 8821  βŸ¨β€œcs1 14545   Ξ£g cgsu 17386  freeMndcfrmd 18728  mCNcmcn 34451  mVRcmvar 34452  mRExcmrex 34457  mRSubstcmrsub 34461
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-pm 8823  df-mrsub 34481
This theorem is referenced by:  mrsubval  34500  mrsubrn  34504  elmrsubrn  34511
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