Theorem ov6g 7510

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 7349	. 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2		eqid 2731	. . . . . 6 ⊢ 𝑆 = 𝑆
3		biidd 262	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑆 = 𝑆 ↔ 𝑆 = 𝑆))
4	3	copsex2g 5431	. . . . . 6 ⊢ ((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐻) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑆 = 𝑆) ↔ 𝑆 = 𝑆))
5	2, 4	mpbiri 258	. . . . 5 ⊢ ((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐻) → ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑆 = 𝑆))
6	5	3adant3 1132	. . . 4 ⊢ ((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐻 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) → ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑆 = 𝑆))
7	6	adantr 480	. . 3 ⊢ (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐻 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) ∧ 𝑆 ∈ 𝐽) → ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑆 = 𝑆))
8		eqeq1 2735	. . . . . . . 8 ⊢ (𝑤 = ⟨𝐴, 𝐵⟩ → (𝑤 = ⟨𝑥, 𝑦⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩))
9	8	anbi1d 631	. . . . . . 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 2742	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ → (𝑧 = 𝑅 ↔ 𝑧 = 𝑆))
12	11	eqcoms 2739	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → (𝑧 = 𝑅 ↔ 𝑧 = 𝑆))
13	12	pm5.32i 574	. . . . . . 7 ⊢ ((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑆))
14	9, 13	bitrdi 287	. . . . . 6 ⊢ (𝑤 = ⟨𝐴, 𝐵⟩ → ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑆)))
15	14	2exbidv 1925	. . . . 5 ⊢ (𝑤 = ⟨𝐴, 𝐵⟩ → (∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅) ↔ ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑆)))
16		eqeq1 2735	. . . . . . 7 ⊢ (𝑧 = 𝑆 → (𝑧 = 𝑆 ↔ 𝑆 = 𝑆))
17	16	anbi2d 630	. . . . . 6 ⊢ (𝑧 = 𝑆 → ((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑆) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑆 = 𝑆)))
18	17	2exbidv 1925	. . . . 5 ⊢ (𝑧 = 𝑆 → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑆) ↔ ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑆 = 𝑆)))
19		moeq 3661	. . . . . . 7 ⊢ ∃*𝑧 𝑧 = 𝑅
20	19	mosubop 5449	. . . . . 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 7404	. . . . . 6 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ 𝑧 = 𝑅)} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ 𝑧 = 𝑅))}
24		eleq1 2819	. . . . . . . . . . . 12 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤 ∈ 𝐶 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐶))
25	24	anbi1d 631	. . . . . . . . . . 11 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝑤 ∈ 𝐶 ∧ 𝑧 = 𝑅) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ 𝑧 = 𝑅)))
26	25	pm5.32i 574	. . . . . . . . . 10 ⊢ ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑤 ∈ 𝐶 ∧ 𝑧 = 𝑅)) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ 𝑧 = 𝑅)))
27		an12 645	. . . . . . . . . 10 ⊢ ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝑤 ∈ 𝐶 ∧ 𝑧 = 𝑅)) ↔ (𝑤 ∈ 𝐶 ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅)))
28	26, 27	bitr3i 277	. . . . . . . . 9 ⊢ ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ 𝑧 = 𝑅)) ↔ (𝑤 ∈ 𝐶 ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅)))
29	28	2exbii 1850	. . . . . . . 8 ⊢ (∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ 𝑧 = 𝑅)) ↔ ∃𝑥∃𝑦(𝑤 ∈ 𝐶 ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅)))
30		19.42vv 1958	. . . . . . . 8 ⊢ (∃𝑥∃𝑦(𝑤 ∈ 𝐶 ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅)) ↔ (𝑤 ∈ 𝐶 ∧ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅)))
31	29, 30	bitri 275	. . . . . . 7 ⊢ (∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ 𝑧 = 𝑅)) ↔ (𝑤 ∈ 𝐶 ∧ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅)))
32	31	opabbii 5156	. . . . . 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 2758	. . . . 5 ⊢ 𝐹 = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝐶 ∧ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑧 = 𝑅))}
34	15, 18, 21, 33	fvopab3ig 6925	. . . 4 ⊢ ((⟨𝐴, 𝐵⟩ ∈ 𝐶 ∧ 𝑆 ∈ 𝐽) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑆 = 𝑆) → (𝐹‘⟨𝐴, 𝐵⟩) = 𝑆))
35	34	3ad2antl3 1188	. . 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 15	. 2 ⊢ (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐻 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) ∧ 𝑆 ∈ 𝐽) → (𝐹‘⟨𝐴, 𝐵⟩) = 𝑆)
37	1, 36	eqtrid 2778	1 ⊢ (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐻 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) ∧ 𝑆 ∈ 𝐽) → (𝐴𝐹𝐵) = 𝑆)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2111  ∃*wmo 2533  ⟨cop 4579  {copab 5151  ‘cfv 6481  (class class class)co 7346  {coprab 7347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-iota 6437  df-fun 6483  df-fv 6489  df-ov 7349  df-oprab 7350

This theorem is referenced by: (None)