Theorem ovidig 7498

Description: The value of an operation class abstraction. Compare ovidi 7499. The condition (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) is been removed. (Contributed by Mario Carneiro, 29-Dec-2014.)

Hypotheses

Ref	Expression
ovidig.1	⊢ ∃*𝑧𝜑
ovidig.2	⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}

Assertion

Ref	Expression
ovidig	⊢ (𝜑 → (𝑥𝐹𝑦) = 𝑧)

Distinct variable group:   𝑥,𝑦,𝑧

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

Proof of Theorem ovidig

Step	Hyp	Ref	Expression
1		df-ov 7359	. 2 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
2		ovidig.1	. . . . 5 ⊢ ∃*𝑧𝜑
3	2	funoprab 7478	. . . 4 ⊢ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
4		ovidig.2	. . . . 5 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
5	4	funeqi 6511	. . . 4 ⊢ (Fun 𝐹 ↔ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
6	3, 5	mpbir 231	. . 3 ⊢ Fun 𝐹
7		oprabidw 7387	. . . . 5 ⊢ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜑)
8	7	biimpri 228	. . . 4 ⊢ (𝜑 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
9	8, 4	eleqtrrdi 2845	. . 3 ⊢ (𝜑 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹)
10		funopfv 6881	. . 3 ⊢ (Fun 𝐹 → (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹 → (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧))
11	6, 9, 10	mpsyl 68	. 2 ⊢ (𝜑 → (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)
12	1, 11	eqtrid 2781	1 ⊢ (𝜑 → (𝑥𝐹𝑦) = 𝑧)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ∃*wmo 2535  ⟨cop 4584  Fun wfun 6484  ‘cfv 6490  (class class class)co 7356  {coprab 7357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-iota 6446  df-fun 6492  df-fv 6498  df-ov 7359  df-oprab 7360

This theorem is referenced by:  ovidi  7499