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Theorem mrsubffval 34567
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
mrsubffval (𝑇 ∈ π‘Š 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)))))
Distinct variable groups:   𝑒,𝑓,𝑣,𝐢   𝑅,𝑒,𝑓,𝑣   𝑒,𝐺,𝑓   𝑇,𝑒,𝑓,𝑣   𝑒,𝑉,𝑓,𝑣
Allowed substitution hints:   𝑆(𝑣,𝑒,𝑓)   𝐺(𝑣)   π‘Š(𝑣,𝑒,𝑓)

Proof of Theorem mrsubffval
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 mrsubffval.s . 2 𝑆 = (mRSubstβ€˜π‘‡)
2 elex 3492 . . 3 (𝑇 ∈ π‘Š β†’ 𝑇 ∈ V)
3 fveq2 6891 . . . . . . 7 (𝑑 = 𝑇 β†’ (mRExβ€˜π‘‘) = (mRExβ€˜π‘‡))
4 mrsubffval.r . . . . . . 7 𝑅 = (mRExβ€˜π‘‡)
53, 4eqtr4di 2790 . . . . . 6 (𝑑 = 𝑇 β†’ (mRExβ€˜π‘‘) = 𝑅)
6 fveq2 6891 . . . . . . 7 (𝑑 = 𝑇 β†’ (mVRβ€˜π‘‘) = (mVRβ€˜π‘‡))
7 mrsubffval.v . . . . . . 7 𝑉 = (mVRβ€˜π‘‡)
86, 7eqtr4di 2790 . . . . . 6 (𝑑 = 𝑇 β†’ (mVRβ€˜π‘‘) = 𝑉)
95, 8oveq12d 7429 . . . . 5 (𝑑 = 𝑇 β†’ ((mRExβ€˜π‘‘) ↑pm (mVRβ€˜π‘‘)) = (𝑅 ↑pm 𝑉))
10 fveq2 6891 . . . . . . . . . . 11 (𝑑 = 𝑇 β†’ (mCNβ€˜π‘‘) = (mCNβ€˜π‘‡))
11 mrsubffval.c . . . . . . . . . . 11 𝐢 = (mCNβ€˜π‘‡)
1210, 11eqtr4di 2790 . . . . . . . . . 10 (𝑑 = 𝑇 β†’ (mCNβ€˜π‘‘) = 𝐢)
1312, 8uneq12d 4164 . . . . . . . . 9 (𝑑 = 𝑇 β†’ ((mCNβ€˜π‘‘) βˆͺ (mVRβ€˜π‘‘)) = (𝐢 βˆͺ 𝑉))
1413fveq2d 6895 . . . . . . . 8 (𝑑 = 𝑇 β†’ (freeMndβ€˜((mCNβ€˜π‘‘) βˆͺ (mVRβ€˜π‘‘))) = (freeMndβ€˜(𝐢 βˆͺ 𝑉)))
15 mrsubffval.g . . . . . . . 8 𝐺 = (freeMndβ€˜(𝐢 βˆͺ 𝑉))
1614, 15eqtr4di 2790 . . . . . . 7 (𝑑 = 𝑇 β†’ (freeMndβ€˜((mCNβ€˜π‘‘) βˆͺ (mVRβ€˜π‘‘))) = 𝐺)
1713mpteq1d 5243 . . . . . . . 8 (𝑑 = 𝑇 β†’ (𝑣 ∈ ((mCNβ€˜π‘‘) βˆͺ (mVRβ€˜π‘‘)) ↦ if(𝑣 ∈ dom 𝑓, (π‘“β€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) = (𝑣 ∈ (𝐢 βˆͺ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (π‘“β€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)))
1817coeq1d 5861 . . . . . . 7 (𝑑 = 𝑇 β†’ ((𝑣 ∈ ((mCNβ€˜π‘‘) βˆͺ (mVRβ€˜π‘‘)) ↦ if(𝑣 ∈ dom 𝑓, (π‘“β€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒) = ((𝑣 ∈ (𝐢 βˆͺ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (π‘“β€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒))
1916, 18oveq12d 7429 . . . . . 6 (𝑑 = 𝑇 β†’ ((freeMndβ€˜((mCNβ€˜π‘‘) βˆͺ (mVRβ€˜π‘‘))) Ξ£g ((𝑣 ∈ ((mCNβ€˜π‘‘) βˆͺ (mVRβ€˜π‘‘)) ↦ if(𝑣 ∈ dom 𝑓, (π‘“β€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒)) = (𝐺 Ξ£g ((𝑣 ∈ (𝐢 βˆͺ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (π‘“β€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒)))
205, 19mpteq12dv 5239 . . . . 5 (𝑑 = 𝑇 β†’ (𝑒 ∈ (mRExβ€˜π‘‘) ↦ ((freeMndβ€˜((mCNβ€˜π‘‘) βˆͺ (mVRβ€˜π‘‘))) Ξ£g ((𝑣 ∈ ((mCNβ€˜π‘‘) βˆͺ (mVRβ€˜π‘‘)) ↦ if(𝑣 ∈ dom 𝑓, (π‘“β€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒))) = (𝑒 ∈ 𝑅 ↦ (𝐺 Ξ£g ((𝑣 ∈ (𝐢 βˆͺ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (π‘“β€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒))))
219, 20mpteq12dv 5239 . . . 4 (𝑑 = 𝑇 β†’ (𝑓 ∈ ((mRExβ€˜π‘‘) ↑pm (mVRβ€˜π‘‘)) ↦ (𝑒 ∈ (mRExβ€˜π‘‘) ↦ ((freeMndβ€˜((mCNβ€˜π‘‘) βˆͺ (mVRβ€˜π‘‘))) Ξ£g ((𝑣 ∈ ((mCNβ€˜π‘‘) βˆͺ (mVRβ€˜π‘‘)) ↦ if(𝑣 ∈ dom 𝑓, (π‘“β€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒)))) = (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)))))
22 df-mrsub 34550 . . . 4 mRSubst = (𝑑 ∈ V ↦ (𝑓 ∈ ((mRExβ€˜π‘‘) ↑pm (mVRβ€˜π‘‘)) ↦ (𝑒 ∈ (mRExβ€˜π‘‘) ↦ ((freeMndβ€˜((mCNβ€˜π‘‘) βˆͺ (mVRβ€˜π‘‘))) Ξ£g ((𝑣 ∈ ((mCNβ€˜π‘‘) βˆͺ (mVRβ€˜π‘‘)) ↦ if(𝑣 ∈ dom 𝑓, (π‘“β€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒)))))
23 ovex 7444 . . . . 5 (𝑅 ↑pm 𝑉) ∈ V
2423mptex 7227 . . . 4 (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)))) ∈ V
2521, 22, 24fvmpt 6998 . . 3 (𝑇 ∈ V β†’ (mRSubstβ€˜π‘‡) = (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)))))
262, 25syl 17 . 2 (𝑇 ∈ π‘Š β†’ (mRSubstβ€˜π‘‡) = (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)))))
271, 26eqtrid 2784 1 (𝑇 ∈ π‘Š 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)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  Vcvv 3474   βˆͺ cun 3946  ifcif 4528   ↦ cmpt 5231  dom cdm 5676   ∘ ccom 5680  β€˜cfv 6543  (class class class)co 7411   ↑pm cpm 8823  βŸ¨β€œcs1 14547   Ξ£g cgsu 17388  freeMndcfrmd 18730  mCNcmcn 34520  mVRcmvar 34521  mRExcmrex 34526  mRSubstcmrsub 34530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-mrsub 34550
This theorem is referenced by:  mrsubfval  34568  mrsubff  34572
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