Theorem ovidig 6654

Description: The value of an operation class abstraction. Compare ovidi 6655. 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 6530	. 2 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
2		ovidig.1	. . . . 5 ⊢ ∃*𝑧𝜑
3	2	funoprab 6636	. . . 4 ⊢ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
4		ovidig.2	. . . . 5 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
5	4	funeqi 5810	. . . 4 ⊢ (Fun 𝐹 ↔ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
6	3, 5	mpbir 219	. . 3 ⊢ Fun 𝐹
7		oprabid 6554	. . . . 5 ⊢ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜑)
8	7	biimpri 216	. . . 4 ⊢ (𝜑 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
9	8, 4	syl6eleqr 2698	. . 3 ⊢ (𝜑 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹)
10		funopfv 6130	. . 3 ⊢ (Fun 𝐹 → (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹 → (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧))
11	6, 9, 10	mpsyl 65	. 2 ⊢ (𝜑 → (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)
12	1, 11	syl5eq 2655	1 ⊢ (𝜑 → (𝑥𝐹𝑦) = 𝑧)

Syntax hints:   → wi 4   = wceq 1474   ∈ wcel 1976  ∃*wmo 2458  ⟨cop 4130  Fun wfun 5784  ‘cfv 5790  (class class class)co 6527  {coprab 6528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828

This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fv 5798  df-ov 6530  df-oprab 6531

This theorem is referenced by:  ovidi  6655