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Theorem mvrval 19469
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)
Assertion
Ref Expression
mvrval (𝜑 → (𝑉𝑋) = (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )))
Distinct variable groups:   0 ,𝑓   1 ,𝑓   𝑦,𝑓,𝐷   𝑦,𝑊   𝑓,,𝐼,𝑦   𝑅,𝑓   𝑓,𝑋,,𝑦
Allowed substitution hints:   𝜑(𝑦,𝑓,)   𝐷()   𝑅(𝑦,)   1 (𝑦,)   𝑉(𝑦,𝑓,)   𝑊(𝑓,)   𝑌(𝑦,𝑓,)   0 (𝑦,)

Proof of Theorem mvrval
Dummy variable 𝑥 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)
71, 2, 3, 4, 5, 6mvrfval 19468 . . 3 (𝜑𝑉 = (𝑥𝐼 ↦ (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))))
87fveq1d 6231 . 2 (𝜑 → (𝑉𝑋) = ((𝑥𝐼 ↦ (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 )))‘𝑋))
9 mvrval.x . . 3 (𝜑𝑋𝐼)
10 eqeq2 2662 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑦 = 𝑥𝑦 = 𝑋))
1110ifbid 4141 . . . . . . . 8 (𝑥 = 𝑋 → if(𝑦 = 𝑥, 1, 0) = if(𝑦 = 𝑋, 1, 0))
1211mpteq2dv 4778 . . . . . . 7 (𝑥 = 𝑋 → (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)) = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))
1312eqeq2d 2661 . . . . . 6 (𝑥 = 𝑋 → (𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)) ↔ 𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
1413ifbid 4141 . . . . 5 (𝑥 = 𝑋 → if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ) = if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))
1514mpteq2dv 4778 . . . 4 (𝑥 = 𝑋 → (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 )) = (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )))
16 eqid 2651 . . . 4 (𝑥𝐼 ↦ (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))) = (𝑥𝐼 ↦ (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 )))
17 ovex 6718 . . . . . 6 (ℕ0𝑚 𝐼) ∈ V
182, 17rabex2 4847 . . . . 5 𝐷 ∈ V
1918mptex 6527 . . . 4 (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )) ∈ V
2015, 16, 19fvmpt 6321 . . 3 (𝑋𝐼 → ((𝑥𝐼 ↦ (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 )))‘𝑋) = (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )))
219, 20syl 17 . 2 (𝜑 → ((𝑥𝐼 ↦ (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 )))‘𝑋) = (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )))
228, 21eqtrd 2685 1 (𝜑 → (𝑉𝑋) = (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  {crab 2945  ifcif 4119  cmpt 4762  ccnv 5142  cima 5146  cfv 5926  (class class class)co 6690  𝑚 cmap 7899  Fincfn 7997  0cc0 9974  1c1 9975  cn 11058  0cn0 11330  0gc0g 16147  1rcur 18547   mVar cmvr 19400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-mvr 19405
This theorem is referenced by:  mvrval2  19470  mplcoe3  19514  evlslem1  19563
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