Theorem dfxp2 5114

Description: Define cross product via the set construction functions. (Contributed by SF, 8-Jan-2015.)

Assertion

Ref	Expression
dfxp2	⊢ (A × B) = ((◡1st “ A) ∩ (◡2nd “ B))

Proof of Theorem dfxp2

Dummy variables x y z w v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eeanv 1913	. . . . 5 ⊢ (∃w∃v(x = ⟨y, w⟩ ∧ x = ⟨v, z⟩) ↔ (∃w x = ⟨y, w⟩ ∧ ∃v x = ⟨v, z⟩))
2		vex 2863	. . . . . . . 8 ⊢ z ∈ V
3		vex 2863	. . . . . . . 8 ⊢ y ∈ V
4		opeq2 4580	. . . . . . . . . 10 ⊢ (w = z → ⟨y, w⟩ = ⟨y, z⟩)
5	4	eqeq2d 2364	. . . . . . . . 9 ⊢ (w = z → (x = ⟨y, w⟩ ↔ x = ⟨y, z⟩))
6		opeq1 4579	. . . . . . . . . 10 ⊢ (v = y → ⟨v, z⟩ = ⟨y, z⟩)
7	6	eqeq2d 2364	. . . . . . . . 9 ⊢ (v = y → (x = ⟨v, z⟩ ↔ x = ⟨y, z⟩))
8	5, 7	bi2anan9 843	. . . . . . . 8 ⊢ ((w = z ∧ v = y) → ((x = ⟨y, w⟩ ∧ x = ⟨v, z⟩) ↔ (x = ⟨y, z⟩ ∧ x = ⟨y, z⟩)))
9	2, 3, 8	spc2ev 2948	. . . . . . 7 ⊢ ((x = ⟨y, z⟩ ∧ x = ⟨y, z⟩) → ∃w∃v(x = ⟨y, w⟩ ∧ x = ⟨v, z⟩))
10	9	anidms 626	. . . . . 6 ⊢ (x = ⟨y, z⟩ → ∃w∃v(x = ⟨y, w⟩ ∧ x = ⟨v, z⟩))
11		simpl 443	. . . . . . . 8 ⊢ ((x = ⟨y, w⟩ ∧ x = ⟨v, z⟩) → x = ⟨y, w⟩)
12		eqtr2 2371	. . . . . . . . 9 ⊢ ((x = ⟨y, w⟩ ∧ x = ⟨v, z⟩) → ⟨y, w⟩ = ⟨v, z⟩)
13		opth 4603	. . . . . . . . . 10 ⊢ (⟨y, w⟩ = ⟨v, z⟩ ↔ (y = v ∧ w = z))
14	4	adantl 452	. . . . . . . . . 10 ⊢ ((y = v ∧ w = z) → ⟨y, w⟩ = ⟨y, z⟩)
15	13, 14	sylbi 187	. . . . . . . . 9 ⊢ (⟨y, w⟩ = ⟨v, z⟩ → ⟨y, w⟩ = ⟨y, z⟩)
16	12, 15	syl 15	. . . . . . . 8 ⊢ ((x = ⟨y, w⟩ ∧ x = ⟨v, z⟩) → ⟨y, w⟩ = ⟨y, z⟩)
17	11, 16	eqtrd 2385	. . . . . . 7 ⊢ ((x = ⟨y, w⟩ ∧ x = ⟨v, z⟩) → x = ⟨y, z⟩)
18	17	exlimivv 1635	. . . . . 6 ⊢ (∃w∃v(x = ⟨y, w⟩ ∧ x = ⟨v, z⟩) → x = ⟨y, z⟩)
19	10, 18	impbii 180	. . . . 5 ⊢ (x = ⟨y, z⟩ ↔ ∃w∃v(x = ⟨y, w⟩ ∧ x = ⟨v, z⟩))
20		brcnv 4893	. . . . . . 7 ⊢ (y◡1st x ↔ x1st y)
21	3	br1st 4859	. . . . . . 7 ⊢ (x1st y ↔ ∃w x = ⟨y, w⟩)
22	20, 21	bitri 240	. . . . . 6 ⊢ (y◡1st x ↔ ∃w x = ⟨y, w⟩)
23		brcnv 4893	. . . . . . 7 ⊢ (z◡2nd x ↔ x2nd z)
24	2	br2nd 4860	. . . . . . 7 ⊢ (x2nd z ↔ ∃v x = ⟨v, z⟩)
25	23, 24	bitri 240	. . . . . 6 ⊢ (z◡2nd x ↔ ∃v x = ⟨v, z⟩)
26	22, 25	anbi12i 678	. . . . 5 ⊢ ((y◡1st x ∧ z◡2nd x) ↔ (∃w x = ⟨y, w⟩ ∧ ∃v x = ⟨v, z⟩))
27	1, 19, 26	3bitr4i 268	. . . 4 ⊢ (x = ⟨y, z⟩ ↔ (y◡1st x ∧ z◡2nd x))
28	27	2rexbii 2642	. . 3 ⊢ (∃y ∈ A ∃z ∈ B x = ⟨y, z⟩ ↔ ∃y ∈ A ∃z ∈ B (y◡1st x ∧ z◡2nd x))
29		elxp2 4803	. . 3 ⊢ (x ∈ (A × B) ↔ ∃y ∈ A ∃z ∈ B x = ⟨y, z⟩)
30		elima 4755	. . . . 5 ⊢ (x ∈ (◡1st “ A) ↔ ∃y ∈ A y◡1st x)
31		elima 4755	. . . . 5 ⊢ (x ∈ (◡2nd “ B) ↔ ∃z ∈ B z◡2nd x)
32	30, 31	anbi12i 678	. . . 4 ⊢ ((x ∈ (◡1st “ A) ∧ x ∈ (◡2nd “ B)) ↔ (∃y ∈ A y◡1st x ∧ ∃z ∈ B z◡2nd x))
33		elin 3220	. . . 4 ⊢ (x ∈ ((◡1st “ A) ∩ (◡2nd “ B)) ↔ (x ∈ (◡1st “ A) ∧ x ∈ (◡2nd “ B)))
34		reeanv 2779	. . . 4 ⊢ (∃y ∈ A ∃z ∈ B (y◡1st x ∧ z◡2nd x) ↔ (∃y ∈ A y◡1st x ∧ ∃z ∈ B z◡2nd x))
35	32, 33, 34	3bitr4i 268	. . 3 ⊢ (x ∈ ((◡1st “ A) ∩ (◡2nd “ B)) ↔ ∃y ∈ A ∃z ∈ B (y◡1st x ∧ z◡2nd x))
36	28, 29, 35	3bitr4i 268	. 2 ⊢ (x ∈ (A × B) ↔ x ∈ ((◡1st “ A) ∩ (◡2nd “ B)))
37	36	eqriv 2350	1 ⊢ (A × B) = ((◡1st “ A) ∩ (◡2nd “ B))

Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2616   ∩ cin 3209  ⟨cop 4562   class class class wbr 4640  1^st c1st 4718   “ cima 4723   × cxp 4771  ^◡ccnv 4772  2^nd c2nd 4784

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-1st 4724  df-ima 4728  df-xp 4785  df-cnv 4786  df-2nd 4798

This theorem is referenced by:  xpexg  5115