Theorem aovmpt4g 47795

Description: Value of a function given by the maps-to notation, analogous to ovmpt4g 7543. (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 8055	. . . . . 6 ⊢ (𝐶 ∈ 𝑉 → dom 𝐹 = (𝐴 × 𝐵))
3		opelxpi 5684	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
4		eleq2 2851	. . . . . . 7 ⊢ (dom 𝐹 = (𝐴 × 𝐵) → (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
5	3, 4	imbitrrid 248	. . . . . 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 411	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝐶 ∈ 𝑉) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
8	7	3impa 1122	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
9	1	mpofun 7520	. . . 4 ⊢ Fun 𝐹
10		funres 6563	. . . 4 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ {⟨𝑥, 𝑦⟩}))
11	9, 10	ax-mp 5	. . 3 ⊢ Fun (𝐹 ↾ {⟨𝑥, 𝑦⟩})
12		df-dfat 47713	. . . 4 ⊢ (𝐹 defAt ⟨𝑥, 𝑦⟩ ↔ (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {⟨𝑥, 𝑦⟩})))
13		aovfundmoveq 47775	. . . 4 ⊢ (𝐹 defAt ⟨𝑥, 𝑦⟩ → ((𝑥𝐹𝑦)) = (𝑥𝐹𝑦))
14	12, 13	sylbir 237	. . 3 ⊢ ((⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {⟨𝑥, 𝑦⟩})) → ((𝑥𝐹𝑦)) = (𝑥𝐹𝑦))
15	8, 11, 14	sylancl 595	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → ((𝑥𝐹𝑦)) = (𝑥𝐹𝑦))
16	1	ovmpt4g 7543	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → (𝑥𝐹𝑦) = 𝐶)
17	15, 16	eqtrd 2797	1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → ((𝑥𝐹𝑦)) = 𝐶)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142  {csn 4582  ⟨cop 4588   × cxp 5645  dom cdm 5647   ↾ cres 5649  Fun wfun 6515  (class class class)co 7396   ∈ cmpo 7398   defAt wdfat 47710   ((caov 47712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-aiota 47679  df-dfat 47713  df-afv 47714  df-aov 47715

This theorem is referenced by: (None)