Theorem ovid 7574

Description: The value of an operation class abstraction. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Hypotheses

Ref	Expression
ovid.1	⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ∃!𝑧𝜑)
ovid.2	⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}

Assertion

Ref	Expression
ovid	⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ((𝑥𝐹𝑦) = 𝑧 ↔ 𝜑))

Distinct variable groups:   𝑥,𝑦,𝑧   𝑧,𝑅   𝑧,𝑆

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

Proof of Theorem ovid

Step	Hyp	Ref	Expression
1		df-ov 7434	. . 3 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
2	1	eqeq1i 2742	. 2 ⊢ ((𝑥𝐹𝑦) = 𝑧 ↔ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)
3		ovid.1	. . . . . 6 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ∃!𝑧𝜑)
4	3	fnoprab 7558	. . . . 5 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)} Fn {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)}
5		ovid.2	. . . . . 6 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}
6	5	fneq1i 6665	. . . . 5 ⊢ (𝐹 Fn {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)} ↔ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)} Fn {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)})
7	4, 6	mpbir 231	. . . 4 ⊢ 𝐹 Fn {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)}
8		opabidw 5529	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)} ↔ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆))
9	8	biimpri 228	. . . 4 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)})
10		fnopfvb 6960	. . . 4 ⊢ ((𝐹 Fn {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)} ∧ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)}) → ((𝐹‘⟨𝑥, 𝑦⟩) = 𝑧 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹))
11	7, 9, 10	sylancr 587	. . 3 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ((𝐹‘⟨𝑥, 𝑦⟩) = 𝑧 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹))
12	5	eleq2i 2833	. . . . 5 ⊢ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)})
13		oprabidw 7462	. . . . 5 ⊢ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)} ↔ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑))
14	12, 13	bitri 275	. . . 4 ⊢ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹 ↔ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑))
15	14	baib 535	. . 3 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹 ↔ 𝜑))
16	11, 15	bitrd 279	. 2 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ((𝐹‘⟨𝑥, 𝑦⟩) = 𝑧 ↔ 𝜑))
17	2, 16	bitrid 283	1 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ((𝑥𝐹𝑦) = 𝑧 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∃!weu 2568  ⟨cop 4632  {copab 5205   Fn wfn 6556  ‘cfv 6561  (class class class)co 7431  {coprab 7432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-ov 7434  df-oprab 7435

This theorem is referenced by: (None)