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Theorem mvrval2 20178
 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 20177 . . 3 (𝜑 → (𝑉𝑋) = (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )))
98fveq1d 6645 . 2 (𝜑 → ((𝑉𝑋)‘𝐹) = ((𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))‘𝐹))
10 mvrval2.f . . 3 (𝜑𝐹𝐷)
11 eqeq1 2825 . . . . 5 (𝑓 = 𝐹 → (𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ↔ 𝐹 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
1211ifbid 4462 . . . 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 2821 . . . 4 (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )) = (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))
144fvexi 6657 . . . . 5 1 ∈ V
153fvexi 6657 . . . . 5 0 ∈ V
1614, 15ifex 4488 . . . 4 if(𝐹 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ) ∈ V
1712, 13, 16fvmpt 6741 . . 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 2856 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 1538   ∈ wcel 2115  {crab 3130  ifcif 4440   ↦ cmpt 5119  ◡ccnv 5527   “ cima 5531  ‘cfv 6328  (class class class)co 7130   ↑m cmap 8381  Fincfn 8484  0cc0 10514  1c1 10515  ℕcn 11615  ℕ0cn0 11875  0gc0g 16692  1rcur 19230   mVar cmvr 20108 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pr 5303 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-mvr 20113 This theorem is referenced by:  mvrid  20179  mvrf1  20181  mvrcl  20205
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