Theorem aovmpt4g 45895

Description: Value of a function given by the maps-to notation, analogous to ovmpt4g 7551. (Contributed by Alexander van der Vekens, 26-May-2017.)

Hypothesis

Ref	Expression
aovmpt4g.3	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Assertion

Ref	Expression
aovmpt4g	⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → ((𝑥𝐹𝑦)) = 𝐶)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝑉,𝑦

Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem aovmpt4g

Step	Hyp	Ref	Expression
1		aovmpt4g.3	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
2	1	dmmpog 8057	. . . . . 6 ⊢ (𝐶 ∈ 𝑉 → dom 𝐹 = (𝐴 × 𝐵))
3		opelxpi 5712	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
4		eleq2 2822	. . . . . . 7 ⊢ (dom 𝐹 = (𝐴 × 𝐵) → (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
5	3, 4	imbitrrid 245	. . . . . 6 ⊢ (dom 𝐹 = (𝐴 × 𝐵) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹))
6	2, 5	syl 17	. . . . 5 ⊢ (𝐶 ∈ 𝑉 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹))
7	6	impcom 408	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝐶 ∈ 𝑉) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
8	7	3impa 1110	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
9	1	mpofun 7528	. . . 4 ⊢ Fun 𝐹
10		funres 6587	. . . 4 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ {⟨𝑥, 𝑦⟩}))
11	9, 10	ax-mp 5	. . 3 ⊢ Fun (𝐹 ↾ {⟨𝑥, 𝑦⟩})
12		df-dfat 45813	. . . 4 ⊢ (𝐹 defAt ⟨𝑥, 𝑦⟩ ↔ (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {⟨𝑥, 𝑦⟩})))
13		aovfundmoveq 45875	. . . 4 ⊢ (𝐹 defAt ⟨𝑥, 𝑦⟩ → ((𝑥𝐹𝑦)) = (𝑥𝐹𝑦))
14	12, 13	sylbir 234	. . 3 ⊢ ((⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {⟨𝑥, 𝑦⟩})) → ((𝑥𝐹𝑦)) = (𝑥𝐹𝑦))
15	8, 11, 14	sylancl 586	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → ((𝑥𝐹𝑦)) = (𝑥𝐹𝑦))
16	1	ovmpt4g 7551	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → (𝑥𝐹𝑦) = 𝐶)
17	15, 16	eqtrd 2772	1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → ((𝑥𝐹𝑦)) = 𝐶)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {csn 4627  ⟨cop 4633   × cxp 5673  dom cdm 5675   ↾ cres 5677  Fun wfun 6534  (class class class)co 7405   ∈ cmpo 7407   defAt wdfat 45810   ((caov 45812

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-aiota 45779  df-dfat 45813  df-afv 45814  df-aov 45815

This theorem is referenced by: (None)