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Theorem mvrval2 19637
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 19636 . . 3 (𝜑 → (𝑉𝑋) = (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )))
98fveq1d 6335 . 2 (𝜑 → ((𝑉𝑋)‘𝐹) = ((𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))‘𝐹))
10 mvrval2.f . . 3 (𝜑𝐹𝐷)
11 eqeq1 2775 . . . . 5 (𝑓 = 𝐹 → (𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ↔ 𝐹 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
1211ifbid 4248 . . . 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 2771 . . . 4 (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )) = (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))
144fvexi 6345 . . . . 5 1 ∈ V
153fvexi 6345 . . . . 5 0 ∈ V
1614, 15ifex 4296 . . . 4 if(𝐹 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ) ∈ V
1712, 13, 16fvmpt 6426 . . 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 2805 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 1631  wcel 2145  {crab 3065  ifcif 4226  cmpt 4864  ccnv 5249  cima 5253  cfv 6030  (class class class)co 6796  𝑚 cmap 8013  Fincfn 8113  0cc0 10142  1c1 10143  cn 11226  0cn0 11499  0gc0g 16308  1rcur 18709   mVar cmvr 19567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-mvr 19572
This theorem is referenced by:  mvrid  19638  mvrf1  19640  mvrcl  19664
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