Theorem dfxp2 5114

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

Assertion

Ref	Expression
dfxp2	|- (A X. B) = ((`'1st"A) i^i (`'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 |- (E.wE.v(x = <.y, w>. /\ x = <.v, z>.) <-> (E.w x = <.y, w>. /\ E.v x = <.v, z>.))
2		vex 2863	. . . . . . . 8 |- z e. _V
3		vex 2863	. . . . . . . 8 |- y e. _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>.) -> E.wE.v(x = <.y, w>. /\ x = <.v, z>.))
10	9	anidms 626	. . . . . 6 |- (x = <.y, z>. -> E.wE.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 |- (E.wE.v(x = <.y, w>. /\ x = <.v, z>.) -> x = <.y, z>.)
19	10, 18	impbii 180	. . . . 5 |- (x = <.y, z>. <-> E.wE.v(x = <.y, w>. /\ x = <.v, z>.))
20		brcnv 4893	. . . . . . 7 |- (y`'1stx <-> x1sty)
21	3	br1st 4859	. . . . . . 7 |- (x1sty <-> E.w x = <.y, w>.)
22	20, 21	bitri 240	. . . . . 6 |- (y`'1stx <-> E.w x = <.y, w>.)
23		brcnv 4893	. . . . . . 7 |- (z`'2ndx <-> x2ndz)
24	2	br2nd 4860	. . . . . . 7 |- (x2ndz <-> E.v x = <.v, z>.)
25	23, 24	bitri 240	. . . . . 6 |- (z`'2ndx <-> E.v x = <.v, z>.)
26	22, 25	anbi12i 678	. . . . 5 |- ((y`'1stx /\ z`'2ndx) <-> (E.w x = <.y, w>. /\ E.v x = <.v, z>.))
27	1, 19, 26	3bitr4i 268	. . . 4 |- (x = <.y, z>. <-> (y`'1stx /\ z`'2ndx))
28	27	2rexbii 2642	. . 3 |- (E.y e. A E.z e. B x = <.y, z>. <-> E.y e. A E.z e. B (y`'1stx /\ z`'2ndx))
29		elxp2 4803	. . 3 |- (x e. (A X. B) <-> E.y e. A E.z e. B x = <.y, z>.)
30		elima 4755	. . . . 5 |- (x e. (`'1st"A) <-> E.y e. A y`'1stx)
31		elima 4755	. . . . 5 |- (x e. (`'2nd"B) <-> E.z e. B z`'2ndx)
32	30, 31	anbi12i 678	. . . 4 |- ((x e. (`'1st"A) /\ x e. (`'2nd"B)) <-> (E.y e. A y`'1stx /\ E.z e. B z`'2ndx))
33		elin 3220	. . . 4 |- (x e. ((`'1st"A) i^i (`'2nd"B)) <-> (x e. (`'1st"A) /\ x e. (`'2nd"B)))
34		reeanv 2779	. . . 4 |- (E.y e. A E.z e. B (y`'1stx /\ z`'2ndx) <-> (E.y e. A y`'1stx /\ E.z e. B z`'2ndx))
35	32, 33, 34	3bitr4i 268	. . 3 |- (x e. ((`'1st"A) i^i (`'2nd"B)) <-> E.y e. A E.z e. B (y`'1stx /\ z`'2ndx))
36	28, 29, 35	3bitr4i 268	. 2 |- (x e. (A X. B) <-> x e. ((`'1st"A) i^i (`'2nd"B)))
37	36	eqriv 2350	1 |- (A X. B) = ((`'1st"A) i^i (`'2nd"B))

Syntax hints:   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  E.wrex 2616   i^i cin 3209  <.cop 4562   class class class wbr 4640  1stc1st 4718  "cima 4723   X. cxp 4771  `'ccnv 4772  2ndc2nd 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