Theorem ov6g 7575

Description: The value of an operation class abstraction. Special case. (Contributed by NM, 13-Nov-2006.)

Hypotheses

Ref	Expression
ov6g.1	⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ → 𝑅 = 𝑆)
ov6g.2	⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ 𝑧 = 𝑅)}

Assertion

Ref	Expression
ov6g	⊢ (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐻 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) ∧ 𝑆 ∈ 𝐽) → (𝐴𝐹𝐵) = 𝑆)

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

Allowed substitution hints:   𝑅(𝑥,𝑦)   𝐹(𝑥,𝑦,𝑧)   𝐺(𝑥,𝑦,𝑧)   𝐻(𝑥,𝑦,𝑧)   𝐽(𝑥,𝑦,𝑧)

Proof of Theorem ov6g

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ov 7414	. 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2		eqid 2769	. . . . . 6 ⊢ 𝑆 = 𝑆
3		biidd 265	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑆 = 𝑆 ↔ 𝑆 = 𝑆))
4	3	copsex2g 5477	. . . . . 6 ⊢ ((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐻) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑆 = 𝑆) ↔ 𝑆 = 𝑆))
5	2, 4	mpbiri 261	. . . . 5 ⊢ ((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐻) → ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑆 = 𝑆))
6	5	3adant3 1148	. . . 4 ⊢ ((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐻 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) → ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑆 = 𝑆))
7	6	adantr 485	. . 3 ⊢ (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐻 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) ∧ 𝑆 ∈ 𝐽) → ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑆 = 𝑆))
8		eqeq1 2773	. . . . . . . 8 ⊢ (𝑤 = ⟨𝐴, 𝐵⟩ → (𝑤 = ⟨𝑥, 𝑦⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩))
9	8	anbi1d 642	. . . . . . 7 ⊢ (𝑤 = ⟨𝐴, 𝐵⟩ → ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅)))
10		ov6g.1	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ → 𝑅 = 𝑆)
11	10	eqeq2d 2780	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ → (𝑧 = 𝑅 ↔ 𝑧 = 𝑆))
12	11	eqcoms 2777	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → (𝑧 = 𝑅 ↔ 𝑧 = 𝑆))
13	12	pm5.32i 584	. . . . . . 7 ⊢ ((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑆))
14	9, 13	bitrdi 290	. . . . . 6 ⊢ (𝑤 = ⟨𝐴, 𝐵⟩ → ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑆)))
15	14	2exbidv 1951	. . . . 5 ⊢ (𝑤 = ⟨𝐴, 𝐵⟩ → (∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅) ↔ ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑆)))
16		eqeq1 2773	. . . . . . 7 ⊢ (𝑧 = 𝑆 → (𝑧 = 𝑆 ↔ 𝑆 = 𝑆))
17	16	anbi2d 641	. . . . . 6 ⊢ (𝑧 = 𝑆 → ((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑆) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑆 = 𝑆)))
18	17	2exbidv 1951	. . . . 5 ⊢ (𝑧 = 𝑆 → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑆) ↔ ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑆 = 𝑆)))
19		moeq 3679	. . . . . . 7 ⊢ ∃*𝑧 𝑧 = 𝑅
20	19	mosubop 5495	. . . . . 6 ⊢ ∃*𝑧∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅)
21	20	a1i 11	. . . . 5 ⊢ (𝑤 ∈ 𝐶 → ∃*𝑧∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅))
22		ov6g.2	. . . . . 6 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ 𝑧 = 𝑅)}
23		dfoprab2 7469	. . . . . 6 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ 𝑧 = 𝑅)} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ 𝑧 = 𝑅))}
24		eleq1 2857	. . . . . . . . . . . 12 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤 ∈ 𝐶 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐶))
25	24	anbi1d 642	. . . . . . . . . . 11 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝑤 ∈ 𝐶 ∧ 𝑧 = 𝑅) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ 𝑧 = 𝑅)))
26	25	pm5.32i 584	. . . . . . . . . 10 ⊢ ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑤 ∈ 𝐶 ∧ 𝑧 = 𝑅)) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ 𝑧 = 𝑅)))
27		an12 657	. . . . . . . . . 10 ⊢ ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑤 ∈ 𝐶 ∧ 𝑧 = 𝑅)) ↔ (𝑤 ∈ 𝐶 ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅)))
28	26, 27	bitr3i 280	. . . . . . . . 9 ⊢ ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ 𝑧 = 𝑅)) ↔ (𝑤 ∈ 𝐶 ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅)))
29	28	2exbii 1876	. . . . . . . 8 ⊢ (∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ 𝑧 = 𝑅)) ↔ ∃𝑥∃𝑦(𝑤 ∈ 𝐶 ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅)))
30		19.42vv 1984	. . . . . . . 8 ⊢ (∃𝑥∃𝑦(𝑤 ∈ 𝐶 ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅)) ↔ (𝑤 ∈ 𝐶 ∧ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅)))
31	29, 30	bitri 278	. . . . . . 7 ⊢ (∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ 𝑧 = 𝑅)) ↔ (𝑤 ∈ 𝐶 ∧ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅)))
32	31	opabbii 5182	. . . . . 6 ⊢ {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ 𝑧 = 𝑅))} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝐶 ∧ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅))}
33	22, 23, 32	3eqtri 2796	. . . . 5 ⊢ 𝐹 = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝐶 ∧ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅))}
34	15, 18, 21, 33	fvopab3ig 6986	. . . 4 ⊢ ((⟨𝐴, 𝐵⟩ ∈ 𝐶 ∧ 𝑆 ∈ 𝐽) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑆 = 𝑆) → (𝐹‘⟨𝐴, 𝐵⟩) = 𝑆))
35	34	3ad2antl3 1204	. . 3 ⊢ (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐻 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) ∧ 𝑆 ∈ 𝐽) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑆 = 𝑆) → (𝐹‘⟨𝐴, 𝐵⟩) = 𝑆))
36	7, 35	mpd 16	. 2 ⊢ (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐻 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) ∧ 𝑆 ∈ 𝐽) → (𝐹‘⟨𝐴, 𝐵⟩) = 𝑆)
37	1, 36	eqtrid 2816	1 ⊢ (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐻 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) ∧ 𝑆 ∈ 𝐽) → (𝐴𝐹𝐵) = 𝑆)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567  ∃wex 1806   ∈ wcel 2149  ∃*wmo 2571  ⟨cop 4600  {copab 5177  ‘cfv 6537  (class class class)co 7411  {coprab 7412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7414  df-oprab 7415

This theorem is referenced by: (None)