Theorem ovigg 5935

Description: The value of an operation class abstraction. Compare ovig 5936. 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 5902	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜓))
3		df-ov 5821	. . . 4 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
4		ovigg.5	. . . . 5 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
5	4	fveq1i 5466	. . . 4 ⊢ (𝐹‘⟨𝐴, 𝐵⟩) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}‘⟨𝐴, 𝐵⟩)
6	3, 5	eqtri 2178	. . 3 ⊢ (𝐴𝐹𝐵) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}‘⟨𝐴, 𝐵⟩)
7		ovigg.4	. . . . 5 ⊢ ∃*𝑧𝜑
8	7	funoprab 5915	. . . 4 ⊢ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
9		funopfv 5505	. . . 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	syl5eq 2202	. 2 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → (𝐴𝐹𝐵) = 𝐶)
12	2, 11	syl6bir 163	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝜓 → (𝐴𝐹𝐵) = 𝐶))

Syntax hints:   → wi 4   ↔ wb 104   ∧ w3a 963   = wceq 1335  ∃*wmo 2007   ∈ wcel 2128  ⟨cop 3563  Fun wfun 5161  ‘cfv 5167  (class class class)co 5818  {coprab 5819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4252  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-iota 5132  df-fun 5169  df-fv 5175  df-ov 5821  df-oprab 5822

This theorem is referenced by:  ovig  5936