Theorem ov 6780

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

Hypotheses

Ref	Expression
ov.1	⊢ 𝐶 ∈ V
ov.2	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
ov.3	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
ov.4	⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))
ov.5	⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ∃!𝑧𝜑)
ov.6	⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}

Assertion

Ref	Expression
ov	⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵) = 𝐶 ↔ 𝜃))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝜃,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)   𝜒(𝑥,𝑦,𝑧)   𝐹(𝑥,𝑦,𝑧)

Proof of Theorem ov

Step	Hyp	Ref	Expression
1		df-ov 6653	. . . . 5 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2		ov.6	. . . . . 6 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}
3	2	fveq1i 6192	. . . . 5 ⊢ (𝐹‘⟨𝐴, 𝐵⟩) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩)
4	1, 3	eqtri 2644	. . . 4 ⊢ (𝐴𝐹𝐵) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩)
5	4	eqeq1i 2627	. . 3 ⊢ ((𝐴𝐹𝐵) = 𝐶 ↔ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩) = 𝐶)
6		ov.5	. . . . . 6 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ∃!𝑧𝜑)
7	6	fnoprab 6763	. . . . 5 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)} Fn {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)}
8		eleq1 2689	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑅 ↔ 𝐴 ∈ 𝑅))
9	8	anbi1d 741	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ↔ (𝐴 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)))
10		eleq1 2689	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝑆 ↔ 𝐵 ∈ 𝑆))
11	10	anbi2d 740	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ↔ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)))
12	9, 11	opelopabg 4993	. . . . . 6 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)} ↔ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)))
13	12	ibir 257	. . . . 5 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)})
14		fnopfvb 6237	. . . . 5 ⊢ (({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)} Fn {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)} ∧ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)}) → (({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}))
15	7, 13, 14	sylancr 695	. . . 4 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}))
16		ov.1	. . . . 5 ⊢ 𝐶 ∈ V
17		ov.2	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
18	9, 17	anbi12d 747	. . . . . 6 ⊢ (𝑥 = 𝐴 → (((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑) ↔ ((𝐴 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜓)))
19		ov.3	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
20	11, 19	anbi12d 747	. . . . . 6 ⊢ (𝑦 = 𝐵 → (((𝐴 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜓) ↔ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ 𝜒)))
21		ov.4	. . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))
22	21	anbi2d 740	. . . . . 6 ⊢ (𝑧 = 𝐶 → (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ 𝜒) ↔ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ 𝜃)))
23	18, 20, 22	eloprabg 6748	. . . . 5 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ V) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)} ↔ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ 𝜃)))
24	16, 23	mp3an3 1413	. . . 4 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)} ↔ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ 𝜃)))
25	15, 24	bitrd 268	. . 3 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ 𝜃)))
26	5, 25	syl5bb 272	. 2 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵) = 𝐶 ↔ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ 𝜃)))
27	26	bianabs 924	1 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵) = 𝐶 ↔ 𝜃))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∃!weu 2470  Vcvv 3200  ⟨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)