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Theorem mrsubfval 34797
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 34796 . . . . 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 479 . . . 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 5902 . . . . . . . . . . 11 (𝑓 = 𝐹 β†’ dom 𝑓 = dom 𝐹)
9 fdm 6725 . . . . . . . . . . . 12 (𝐹:π΄βŸΆπ‘… β†’ dom 𝐹 = 𝐴)
109ad2antrl 724 . . . . . . . . . . 11 ((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) β†’ dom 𝐹 = 𝐴)
118, 10sylan9eqr 2792 . . . . . . . . . 10 (((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) ∧ 𝑓 = 𝐹) β†’ dom 𝑓 = 𝐴)
1211eleq2d 2817 . . . . . . . . 9 (((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) ∧ 𝑓 = 𝐹) β†’ (𝑣 ∈ dom 𝑓 ↔ 𝑣 ∈ 𝐴))
13 simpr 483 . . . . . . . . . 10 (((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) ∧ 𝑓 = 𝐹) β†’ 𝑓 = 𝐹)
1413fveq1d 6892 . . . . . . . . 9 (((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) ∧ 𝑓 = 𝐹) β†’ (π‘“β€˜π‘£) = (πΉβ€˜π‘£))
1512, 14ifbieq1d 4551 . . . . . . . 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 5249 . . . . . . 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 5860 . . . . . 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 7427 . . . . 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 5249 . . . 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 6904 . . . . . 6 𝑅 ∈ V
2120a1i 11 . . . . 5 ((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) β†’ 𝑅 ∈ V)
222fvexi 6904 . . . . . 6 𝑉 ∈ V
2322a1i 11 . . . . 5 ((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) β†’ 𝑉 ∈ V)
24 simprl 767 . . . . 5 ((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) β†’ 𝐹:π΄βŸΆπ‘…)
25 simprr 769 . . . . 5 ((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) β†’ 𝐴 βŠ† 𝑉)
26 elpm2r 8841 . . . . 5 (((𝑅 ∈ V ∧ 𝑉 ∈ V) ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) β†’ 𝐹 ∈ (𝑅 ↑pm 𝑉))
2721, 23, 24, 25, 26syl22anc 835 . . . 4 ((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) β†’ 𝐹 ∈ (𝑅 ↑pm 𝑉))
2820mptex 7226 . . . . 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 7004 . . 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 411 . 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 6934 . . . 4 (βˆ…β€˜πΉ) = βˆ…
33 fvprc 6882 . . . . . 6 (Β¬ 𝑇 ∈ V β†’ (mRSubstβ€˜π‘‡) = βˆ…)
344, 33eqtrid 2782 . . . . 5 (Β¬ 𝑇 ∈ V β†’ 𝑆 = βˆ…)
3534fveq1d 6892 . . . 4 (Β¬ 𝑇 ∈ V β†’ (π‘†β€˜πΉ) = (βˆ…β€˜πΉ))
36 fvprc 6882 . . . . . . 7 (Β¬ 𝑇 ∈ V β†’ (mRExβ€˜π‘‡) = βˆ…)
373, 36eqtrid 2782 . . . . . 6 (Β¬ 𝑇 ∈ V β†’ 𝑅 = βˆ…)
3837mpteq1d 5242 . . . . 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 6691 . . . . 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 2786 . . . 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 2796 . . 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 394   = wceq 1539   ∈ wcel 2104  Vcvv 3472   βˆͺ cun 3945   βŠ† wss 3947  βˆ…c0 4321  ifcif 4527   ↦ cmpt 5230  dom cdm 5675   ∘ ccom 5679  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411   ↑pm cpm 8823  βŸ¨β€œcs1 14549   Ξ£g cgsu 17390  freeMndcfrmd 18764  mCNcmcn 34749  mVRcmvar 34750  mRExcmrex 34755  mRSubstcmrsub 34759
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-pm 8825  df-mrsub 34779
This theorem is referenced by:  mrsubval  34798  mrsubrn  34802  elmrsubrn  34809
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