Theorem ovigg 6152

Description: The value of an operation class abstraction. Compare ovig 6153. The condition (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) is been 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 6118	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜓))
3		df-ov 6031	. . . 4 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
4		ovigg.5	. . . . 5 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
5	4	fveq1i 5649	. . . 4 ⊢ (𝐹‘⟨𝐴, 𝐵⟩) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}‘⟨𝐴, 𝐵⟩)
6	3, 5	eqtri 2252	. . 3 ⊢ (𝐴𝐹𝐵) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}‘⟨𝐴, 𝐵⟩)
7		ovigg.4	. . . . 5 ⊢ ∃*𝑧𝜑
8	7	funoprab 6131	. . . 4 ⊢ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
9		funopfv 5692	. . . 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 2276	. 2 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → (𝐴𝐹𝐵) = 𝐶)
12	2, 11	biimtrrdi 164	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝜓 → (𝐴𝐹𝐵) = 𝐶))

Syntax hints:   → wi 4   ↔ wb 105   ∧ w3a 1005   = wceq 1398  ∃*wmo 2080   ∈ wcel 2202  ⟨cop 3676  Fun wfun 5327  ‘cfv 5333  (class class class)co 6028  {coprab 6029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fv 5341  df-ov 6031  df-oprab 6032

This theorem is referenced by:  ovig  6153