Theorem ovidig 7594

Description: The value of an operation class abstraction. Compare ovidi 7595. 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 7454	. 2 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
2		ovidig.1	. . . . 5 ⊢ ∃*𝑧𝜑
3	2	funoprab 7573	. . . 4 ⊢ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
4		ovidig.2	. . . . 5 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
5	4	funeqi 6588	. . . 4 ⊢ (Fun 𝐹 ↔ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
6	3, 5	mpbir 234	. . 3 ⊢ Fun 𝐹
7		oprabidw 7482	. . . . 5 ⊢ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜑)
8	7	biimpri 231	. . . 4 ⊢ (𝜑 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
9	8, 4	eleqtrrdi 2875	. . 3 ⊢ (𝜑 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹)
10		funopfv 6965	. . 3 ⊢ (Fun 𝐹 → (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹 → (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧))
11	6, 9, 10	mpsyl 69	. 2 ⊢ (𝜑 → (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)
12	1, 11	eqtrid 2810	1 ⊢ (𝜑 → (𝑥𝐹𝑦) = 𝑧)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ∃*wmo 2565  ⟨cop 4840  Fun wfun 6560  ‘cfv 6566  (class class class)co 7451  {coprab 7452

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-iota 6520  df-fun 6568  df-fv 6574  df-ov 7454  df-oprab 7455

This theorem is used by:  ovidi  7595