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Theorem mrsubfval 34166
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 34165 . . . . 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 5863 . . . . . . . . . . 11 (𝑓 = 𝐹 β†’ dom 𝑓 = dom 𝐹)
9 fdm 6681 . . . . . . . . . . . 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 6848 . . . . . . . . 9 (((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) ∧ 𝑓 = 𝐹) β†’ (π‘“β€˜π‘£) = (πΉβ€˜π‘£))
1512, 14ifbieq1d 4514 . . . . . . . 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 5211 . . . . . . 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 5821 . . . . . 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 7377 . . . . 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 5211 . . . 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 6860 . . . . . 6 𝑅 ∈ V
2120a1i 11 . . . . 5 ((𝑇 ∈ V ∧ (𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉)) β†’ 𝑅 ∈ V)
222fvexi 6860 . . . . . 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 8789 . . . . 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 7177 . . . . 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 6959 . . 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 6890 . . . 4 (βˆ…β€˜πΉ) = βˆ…
33 fvprc 6838 . . . . . 6 (Β¬ 𝑇 ∈ V β†’ (mRSubstβ€˜π‘‡) = βˆ…)
344, 33eqtrid 2785 . . . . 5 (Β¬ 𝑇 ∈ V β†’ 𝑆 = βˆ…)
3534fveq1d 6848 . . . 4 (Β¬ 𝑇 ∈ V β†’ (π‘†β€˜πΉ) = (βˆ…β€˜πΉ))
36 fvprc 6838 . . . . . . 7 (Β¬ 𝑇 ∈ V β†’ (mRExβ€˜π‘‡) = βˆ…)
373, 36eqtrid 2785 . . . . . 6 (Β¬ 𝑇 ∈ V β†’ 𝑅 = βˆ…)
3837mpteq1d 5204 . . . . 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 6647 . . . . 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 3447   βˆͺ cun 3912   βŠ† wss 3914  βˆ…c0 4286  ifcif 4490   ↦ cmpt 5192  dom cdm 5637   ∘ ccom 5641  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361   ↑pm cpm 8772  βŸ¨β€œcs1 14492   Ξ£g cgsu 17330  freeMndcfrmd 18665  mCNcmcn 34118  mVRcmvar 34119  mRExcmrex 34124  mRSubstcmrsub 34128
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-mrsub 34148
This theorem is referenced by:  mrsubval  34167  mrsubrn  34171  elmrsubrn  34178
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