Theorem ov6g 5601

Description: The value of an operation class abstraction. Special case. (Contributed by set.mm contributors, 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 5527	. 2 ⊢ (AFB) = (F ‘⟨A, B⟩)
2		eqid 2353	. . . . . 6 ⊢ S = S
3		biidd 228	. . . . . . 7 ⊢ ((x = A ∧ y = B) → (S = S ↔ S = S))
4	3	copsex2g 4610	. . . . . 6 ⊢ ((A ∈ G ∧ B ∈ H) → (∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ S = S) ↔ S = S))
5	2, 4	mpbiri 224	. . . . 5 ⊢ ((A ∈ G ∧ B ∈ H) → ∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ S = S))
6	5	3adant3 975	. . . 4 ⊢ ((A ∈ G ∧ B ∈ H ∧ ⟨A, B⟩ ∈ C) → ∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ S = S))
7	6	adantr 451	. . 3 ⊢ (((A ∈ G ∧ B ∈ H ∧ ⟨A, B⟩ ∈ C) ∧ S ∈ J) → ∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ S = S))
8		eqeq1 2359	. . . . . . . 8 ⊢ (w = ⟨A, B⟩ → (w = ⟨x, y⟩ ↔ ⟨A, B⟩ = ⟨x, y⟩))
9	8	anbi1d 685	. . . . . . 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 2364	. . . . . . . . 9 ⊢ (⟨x, y⟩ = ⟨A, B⟩ → (z = R ↔ z = S))
12	11	eqcoms 2356	. . . . . . . 8 ⊢ (⟨A, B⟩ = ⟨x, y⟩ → (z = R ↔ z = S))
13	12	pm5.32i 618	. . . . . . 7 ⊢ ((⟨A, B⟩ = ⟨x, y⟩ ∧ z = R) ↔ (⟨A, B⟩ = ⟨x, y⟩ ∧ z = S))
14	9, 13	syl6bb 252	. . . . . 6 ⊢ (w = ⟨A, B⟩ → ((w = ⟨x, y⟩ ∧ z = R) ↔ (⟨A, B⟩ = ⟨x, y⟩ ∧ z = S)))
15	14	2exbidv 1628	. . . . 5 ⊢ (w = ⟨A, B⟩ → (∃x∃y(w = ⟨x, y⟩ ∧ z = R) ↔ ∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ z = S)))
16		eqeq1 2359	. . . . . . 7 ⊢ (z = S → (z = S ↔ S = S))
17	16	anbi2d 684	. . . . . 6 ⊢ (z = S → ((⟨A, B⟩ = ⟨x, y⟩ ∧ z = S) ↔ (⟨A, B⟩ = ⟨x, y⟩ ∧ S = S)))
18	17	2exbidv 1628	. . . . 5 ⊢ (z = S → (∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ z = S) ↔ ∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ S = S)))
19		moeq 3013	. . . . . . 7 ⊢ ∃*z z = R
20	19	mosubop 4614	. . . . . 6 ⊢ ∃*z∃x∃y(w = ⟨x, y⟩ ∧ z = R)
21	20	a1i 10	. . . . 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 5559	. . . . . 6 ⊢ {⟨⟨x, y⟩, z⟩ ∣ (⟨x, y⟩ ∈ C ∧ z = R)} = {⟨w, z⟩ ∣ ∃x∃y(w = ⟨x, y⟩ ∧ (⟨x, y⟩ ∈ C ∧ z = R))}
24		eleq1 2413	. . . . . . . . . . . 12 ⊢ (w = ⟨x, y⟩ → (w ∈ C ↔ ⟨x, y⟩ ∈ C))
25	24	anbi1d 685	. . . . . . . . . . 11 ⊢ (w = ⟨x, y⟩ → ((w ∈ C ∧ z = R) ↔ (⟨x, y⟩ ∈ C ∧ z = R)))
26	25	pm5.32i 618	. . . . . . . . . 10 ⊢ ((w = ⟨x, y⟩ ∧ (w ∈ C ∧ z = R)) ↔ (w = ⟨x, y⟩ ∧ (⟨x, y⟩ ∈ C ∧ z = R)))
27		an12 772	. . . . . . . . . 10 ⊢ ((w = ⟨x, y⟩ ∧ (w ∈ C ∧ z = R)) ↔ (w ∈ C ∧ (w = ⟨x, y⟩ ∧ z = R)))
28	26, 27	bitr3i 242	. . . . . . . . 9 ⊢ ((w = ⟨x, y⟩ ∧ (⟨x, y⟩ ∈ C ∧ z = R)) ↔ (w ∈ C ∧ (w = ⟨x, y⟩ ∧ z = R)))
29	28	2exbii 1583	. . . . . . . 8 ⊢ (∃x∃y(w = ⟨x, y⟩ ∧ (⟨x, y⟩ ∈ C ∧ z = R)) ↔ ∃x∃y(w ∈ C ∧ (w = ⟨x, y⟩ ∧ z = R)))
30		19.42vv 1907	. . . . . . . 8 ⊢ (∃x∃y(w ∈ C ∧ (w = ⟨x, y⟩ ∧ z = R)) ↔ (w ∈ C ∧ ∃x∃y(w = ⟨x, y⟩ ∧ z = R)))
31	29, 30	bitri 240	. . . . . . 7 ⊢ (∃x∃y(w = ⟨x, y⟩ ∧ (⟨x, y⟩ ∈ C ∧ z = R)) ↔ (w ∈ C ∧ ∃x∃y(w = ⟨x, y⟩ ∧ z = R)))
32	31	opabbii 4627	. . . . . 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 2377	. . . . 5 ⊢ F = {⟨w, z⟩ ∣ (w ∈ C ∧ ∃x∃y(w = ⟨x, y⟩ ∧ z = R))}
34	15, 18, 21, 33	fvopab3ig 5388	. . . 4 ⊢ ((⟨A, B⟩ ∈ C ∧ S ∈ J) → (∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ S = S) → (F ‘⟨A, B⟩) = S))
35	34	3ad2antl3 1119	. . 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 14	. 2 ⊢ (((A ∈ G ∧ B ∈ H ∧ ⟨A, B⟩ ∈ C) ∧ S ∈ J) → (F ‘⟨A, B⟩) = S)
37	1, 36	syl5eq 2397	1 ⊢ (((A ∈ G ∧ B ∈ H ∧ ⟨A, B⟩ ∈ C) ∧ S ∈ J) → (AFB) = S)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃*wmo 2205  ⟨cop 4562  {copab 4623   ‘cfv 4782  (class class class)co 5526  {coprab 5528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fv 4796  df-ov 5527  df-oprab 5529

This theorem is referenced by: (None)