Theorem ovigg 7545

Description: The value of an operation class abstraction. Compared with ovig 7546, the condition (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) is removed. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 19-Dec-2013.)

Hypotheses

Ref	Expression
ovigg.1	⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓))
ovigg.4	⊢ ∃*𝑧𝜑
ovigg.5	⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}

Assertion

Ref	Expression
ovigg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝜓 → (𝐴𝐹𝐵) = 𝐶))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝜓,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐹(𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)   𝑊(𝑥,𝑦,𝑧)   𝑋(𝑥,𝑦,𝑧)

Proof of Theorem ovigg

Step	Hyp	Ref	Expression
1		ovigg.1	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓))
2	1	eloprabga 7509	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜓))
3		df-ov 7403	. . . 4 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
4		ovigg.5	. . . . 5 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
5	4	fveq1i 6872	. . . 4 ⊢ (𝐹‘⟨𝐴, 𝐵⟩) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}‘⟨𝐴, 𝐵⟩)
6	3, 5	eqtri 2788	. . 3 ⊢ (𝐴𝐹𝐵) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}‘⟨𝐴, 𝐵⟩)
7		ovigg.4	. . . . 5 ⊢ ∃*𝑧𝜑
8	7	funoprab 7522	. . . 4 ⊢ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
9		funopfv 6920	. . . 4 ⊢ (Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}‘⟨𝐴, 𝐵⟩) = 𝐶))
10	8, 9	ax-mp 5	. . 3 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}‘⟨𝐴, 𝐵⟩) = 𝐶)
11	6, 10	eqtrid 2812	. 2 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → (𝐴𝐹𝐵) = 𝐶)
12	2, 11	biimtrrdi 257	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝜓 → (𝐴𝐹𝐵) = 𝐶))

Syntax hints:   → wi 4   ↔ wb 209   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  ∃*wmo 2567  ⟨cop 4591  Fun wfun 6519  ‘cfv 6525  (class class class)co 7400  {coprab 7401

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5105  df-opab 5167  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6481  df-fun 6527  df-fv 6533  df-ov 7403  df-oprab 7404

This theorem is referenced by:  ovig  7546  joinval  18419  meetval  18433