Theorem ovid 6777

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 6653	. . 3 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
2	1	eqeq1i 2627	. 2 ⊢ ((𝑥𝐹𝑦) = 𝑧 ↔ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)
3		ovid.1	. . . . . 6 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ∃!𝑧𝜑)
4	3	fnoprab 6763	. . . . 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 5985	. . . . 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 221	. . . 4 ⊢ 𝐹 Fn {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)}
8		opabid 4982	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)} ↔ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆))
9	8	biimpri 218	. . . 4 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)})
10		fnopfvb 6237	. . . 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 695	. . 3 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ((𝐹‘⟨𝑥, 𝑦⟩) = 𝑧 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹))
12	5	eleq2i 2693	. . . . 5 ⊢ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)})
13		oprabid 6677	. . . . 5 ⊢ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)} ↔ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑))
14	12, 13	bitri 264	. . . 4 ⊢ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹 ↔ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑))
15	14	baib 944	. . 3 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐹 ↔ 𝜑))
16	11, 15	bitrd 268	. 2 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ((𝐹‘⟨𝑥, 𝑦⟩) = 𝑧 ↔ 𝜑))
17	2, 16	syl5bb 272	1 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ((𝑥𝐹𝑦) = 𝑧 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∃!weu 2470  ⟨cop 4183  {copab 4712   Fn wfn 5883  ‘cfv 5888  (class class class)co 6650  {coprab 6651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-ov 6653  df-oprab 6654

This theorem is referenced by: (None)