Theorem mvrval 22002

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 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
mvrfval.z	⊢ 0 = (0g‘𝑅)
mvrfval.o	⊢ 1 = (1r‘𝑅)
mvrfval.i	⊢ (𝜑 → 𝐼 ∈ 𝑊)
mvrfval.r	⊢ (𝜑 → 𝑅 ∈ 𝑌)
mvrval.x	⊢ (𝜑 → 𝑋 ∈ 𝐼)

Assertion

Ref	Expression
mvrval	⊢ (𝜑 → (𝑉‘𝑋) = (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )))

Distinct variable groups:   0 ,𝑓   1 ,𝑓   𝑦,𝑓,𝐷   𝑦,𝑊   𝑓,ℎ,𝐼,𝑦   𝑅,𝑓   𝑓,𝑋,ℎ,𝑦

Allowed substitution hints:   𝜑(𝑦,𝑓,ℎ)   𝐷(ℎ)   𝑅(𝑦,ℎ)   1 (𝑦,ℎ)   𝑉(𝑦,𝑓,ℎ)   𝑊(𝑓,ℎ)   𝑌(𝑦,𝑓,ℎ)   0 (𝑦,ℎ)

Proof of Theorem mvrval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mvrfval.v	. . . 4 ⊢ 𝑉 = (𝐼 mVar 𝑅)
2		mvrfval.d	. . . 4 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
3		mvrfval.z	. . . 4 ⊢ 0 = (0g‘𝑅)
4		mvrfval.o	. . . 4 ⊢ 1 = (1r‘𝑅)
5		mvrfval.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
6		mvrfval.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑌)
7	1, 2, 3, 4, 5, 6	mvrfval 22001	. . 3 ⊢ (𝜑 → 𝑉 = (𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))))
8	7	fveq1d 6908	. 2 ⊢ (𝜑 → (𝑉‘𝑋) = ((𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 )))‘𝑋))
9		mvrval.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐼)
10		eqeq2 2749	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑦 = 𝑥 ↔ 𝑦 = 𝑋))
11	10	ifbid 4549	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → if(𝑦 = 𝑥, 1, 0) = if(𝑦 = 𝑋, 1, 0))
12	11	mpteq2dv 5244	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))
13	12	eqeq2d 2748	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)) ↔ 𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
14	13	ifbid 4549	. . . . 5 ⊢ (𝑥 = 𝑋 → if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ) = if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))
15	14	mpteq2dv 5244	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 )) = (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )))
16		eqid 2737	. . . 4 ⊢ (𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))) = (𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 )))
17		ovex 7464	. . . . . 6 ⊢ (ℕ0 ↑m 𝐼) ∈ V
18	2, 17	rabex2 5341	. . . . 5 ⊢ 𝐷 ∈ V
19	18	mptex 7243	. . . 4 ⊢ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )) ∈ V
20	15, 16, 19	fvmpt 7016	. . 3 ⊢ (𝑋 ∈ 𝐼 → ((𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 )))‘𝑋) = (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )))
21	9, 20	syl 17	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 )))‘𝑋) = (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )))
22	8, 21	eqtrd 2777	1 ⊢ (𝜑 → (𝑉‘𝑋) = (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  {crab 3436  ifcif 4525   ↦ cmpt 5225  ^◡ccnv 5684   “ cima 5688  ‘cfv 6561  (class class class)co 7431   ↑_m cmap 8866  Fincfn 8985  0cc0 11155  1c1 11156  ℕcn 12266  ℕ₀cn0 12526  0_gc0g 17484  1_rcur 20178   mVar cmvr 21925

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-mvr 21930

This theorem is referenced by:  mvrval2  22003  mplcoe3  22056  evlslem1  22106  rhmply1vr1  22391