Theorem mvrval2 20660

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)
mvrval2.f	⊢ (𝜑 → 𝐹 ∈ 𝐷)

Assertion

Ref	Expression
mvrval2	⊢ (𝜑 → ((𝑉‘𝑋)‘𝐹) = if(𝐹 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))

Distinct variable groups:   𝑦,𝐷   𝑦,𝑊   𝑦,ℎ,𝐼   ℎ,𝑋,𝑦

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

Proof of Theorem mvrval2

Dummy variable 𝑓 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		mvrval.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐼)
8	1, 2, 3, 4, 5, 6, 7	mvrval 20659	. . 3 ⊢ (𝜑 → (𝑉‘𝑋) = (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )))
9	8	fveq1d 6647	. 2 ⊢ (𝜑 → ((𝑉‘𝑋)‘𝐹) = ((𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))‘𝐹))
10		mvrval2.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
11		eqeq1 2802	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ↔ 𝐹 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
12	11	ifbid 4447	. . . 4 ⊢ (𝑓 = 𝐹 → if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ) = if(𝐹 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))
13		eqid 2798	. . . 4 ⊢ (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 ))
14	4	fvexi 6659	. . . . 5 ⊢ 1 ∈ V
15	3	fvexi 6659	. . . . 5 ⊢ 0 ∈ V
16	14, 15	ifex 4473	. . . 4 ⊢ if(𝐹 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ) ∈ V
17	12, 13, 16	fvmpt 6745	. . 3 ⊢ (𝐹 ∈ 𝐷 → ((𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))‘𝐹) = if(𝐹 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))
18	10, 17	syl 17	. 2 ⊢ (𝜑 → ((𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))‘𝐹) = if(𝐹 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))
19	9, 18	eqtrd 2833	1 ⊢ (𝜑 → ((𝑉‘𝑋)‘𝐹) = if(𝐹 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  {crab 3110  ifcif 4425   ↦ cmpt 5110  ^◡ccnv 5518   “ cima 5522  ‘cfv 6324  (class class class)co 7135   ↑_m cmap 8389  Fincfn 8492  0cc0 10526  1c1 10527  ℕcn 11625  ℕ₀cn0 11885  0_gc0g 16705  1_rcur 19244   mVar cmvr 20590

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5295

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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-mvr 20595

This theorem is referenced by:  mvrid  20661  mvrf1  20663  mvrcl  20688  mhpvarcl  20798