Theorem mvrval2 22296

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 22295	. . 3 ⊢ (𝜑 → (𝑉‘𝑋) = (𝑓 ∈ 𝐷 ↦ if(𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )))
9	8	fveq1d 6888	. 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 2764	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) ↔ 𝐹 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
12	11	ifbid 4505	. . . 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 2760	. . . 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 6900	. . . . 5 ⊢ 1 ∈ V
15	3	fvexi 6900	. . . . 5 ⊢ 0 ∈ V
16	14, 15	ifex 4532	. . . 4 ⊢ if(𝐹 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ) ∈ V
17	12, 13, 16	fvmpt 6994	. . 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 18	. 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 2795	1 ⊢ (𝜑 → ((𝑉‘𝑋)‘𝐹) = if(𝐹 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  {crab 3412  ifcif 4481   ↦ cmpt 5185  ^◡ccnv 5649   “ cima 5653  ‘cfv 6539  (class class class)co 7421   ↑_m cmap 8851  Fincfn 8977  0cc0 11204  1c1 11205  ℕcn 12339  ℕ₀cn0 12610  0_gc0g 17616  1_rcur 20413   mVar cmvr 22219

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-mvr 22224

This theorem is used by:  mvrid  22297  mvrf1  22299  mvrcl  22305  mhpvarcl  22475  psdmvr  22496  mvrvalind  34217  esplyfval1  34252