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Theorem mvrval2 21414
Description: Value of the generating elements of the power series structure. (Contributed by Mario Carneiro, 7-Jan-2015.)
Hypotheses
Ref Expression
mvrfval.v 𝑉 = (𝐼 mVar 𝑅)
mvrfval.d 𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
mvrfval.z 0 = (0g𝑅)
mvrfval.o 1 = (1r𝑅)
mvrfval.i (𝜑𝐼𝑊)
mvrfval.r (𝜑𝑅𝑌)
mvrval.x (𝜑𝑋𝐼)
mvrval2.f (𝜑𝐹𝐷)
Assertion
Ref Expression
mvrval2 (𝜑 → ((𝑉𝑋)‘𝐹) = if(𝐹 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))
Distinct variable groups:   𝑦,𝐷   𝑦,𝑊   𝑦,,𝐼   ,𝑋,𝑦
Allowed substitution hints:   𝜑(𝑦,)   𝐷()   𝑅(𝑦,)   1 (𝑦,)   𝐹(𝑦,)   𝑉(𝑦,)   𝑊()   𝑌(𝑦,)   0 (𝑦,)

Proof of Theorem mvrval2
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 mvrfval.v . . . 4 𝑉 = (𝐼 mVar 𝑅)
2 mvrfval.d . . . 4 𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
3 mvrfval.z . . . 4 0 = (0g𝑅)
4 mvrfval.o . . . 4 1 = (1r𝑅)
5 mvrfval.i . . . 4 (𝜑𝐼𝑊)
6 mvrfval.r . . . 4 (𝜑𝑅𝑌)
7 mvrval.x . . . 4 (𝜑𝑋𝐼)
81, 2, 3, 4, 5, 6, 7mvrval 21413 . . 3 (𝜑 → (𝑉𝑋) = (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )))
98fveq1d 6848 . 2 (𝜑 → ((𝑉𝑋)‘𝐹) = ((𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))‘𝐹))
10 mvrval2.f . . 3 (𝜑𝐹𝐷)
11 eqeq1 2737 . . . . 5 (𝑓 = 𝐹 → (𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ↔ 𝐹 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
1211ifbid 4513 . . . 4 (𝑓 = 𝐹 → if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ) = if(𝐹 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))
13 eqid 2733 . . . 4 (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )) = (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))
144fvexi 6860 . . . . 5 1 ∈ V
153fvexi 6860 . . . . 5 0 ∈ V
1614, 15ifex 4540 . . . 4 if(𝐹 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ) ∈ V
1712, 13, 16fvmpt 6952 . . 3 (𝐹𝐷 → ((𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))‘𝐹) = if(𝐹 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))
1810, 17syl 17 . 2 (𝜑 → ((𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))‘𝐹) = if(𝐹 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))
199, 18eqtrd 2773 1 (𝜑 → ((𝑉𝑋)‘𝐹) = if(𝐹 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  {crab 3406  ifcif 4490  cmpt 5192  ccnv 5636  cima 5640  cfv 6500  (class class class)co 7361  m cmap 8771  Fincfn 8889  0cc0 11059  1c1 11060  cn 12161  0cn0 12421  0gc0g 17329  1rcur 19921   mVar cmvr 21330
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-pr 5388
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-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-mvr 21335
This theorem is referenced by:  mvrid  21415  mvrf1  21417  mvrcl  21444  mhpvarcl  21561
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