Theorem crossex 5851

Description: The function mapping x and y to their cross product is a set. (Contributed by SF, 11-Feb-2015.)

Assertion

Ref	Expression
crossex	|- Cross e. _V

Proof of Theorem crossex

Dummy variables a b x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-cross 5765	. . 3 |- Cross = (x e. _V, y e. _V |-> (x X. y))
2		rexcom 2773	. . . . 5 |- (E.a e. x E.b e. y z = <.a, b>. <-> E.b e. y E.a e. x z = <.a, b>.)
3		elxp2 4803	. . . . 5 |- (z e. (x X. y) <-> E.a e. x E.b e. y z = <.a, b>.)
4		elin 3220	. . . . . . . 8 |- (<.{b}, <.{z}, <.x, y>.>.>. e. ( Ins2 Ins2 S i^i Ins4 (( Ins2 Ins2 S i^i Ins4 SI3 ( Ins2 `'1st i^i (_V X. `'2nd)))"1c)) <-> (<.{b}, <.{z}, <.x, y>.>.>. e. Ins2 Ins2 S /\ <.{b}, <.{z}, <.x, y>.>.>. e. Ins4 (( Ins2 Ins2 S i^i Ins4 SI3 ( Ins2 `'1st i^i (_V X. `'2nd)))"1c)))
5		snex 4112	. . . . . . . . . . 11 |- {z} e. _V
6	5	otelins2 5792	. . . . . . . . . 10 |- (<.{b}, <.{z}, <.x, y>.>.>. e. Ins2 Ins2 S <-> <.{b}, <.x, y>.>. e. Ins2 S )
7		vex 2863	. . . . . . . . . . 11 |- x e. _V
8	7	otelins2 5792	. . . . . . . . . 10 |- (<.{b}, <.x, y>.>. e. Ins2 S <-> <.{b}, y>. e. S )
9		vex 2863	. . . . . . . . . . 11 |- b e. _V
10		vex 2863	. . . . . . . . . . 11 |- y e. _V
11	9, 10	opelssetsn 4761	. . . . . . . . . 10 |- (<.{b}, y>. e. S <-> b e. y)
12	6, 8, 11	3bitri 262	. . . . . . . . 9 |- (<.{b}, <.{z}, <.x, y>.>.>. e. Ins2 Ins2 S <-> b e. y)
13	10	oqelins4 5795	. . . . . . . . . 10 |- (<.{b}, <.{z}, <.x, y>.>.>. e. Ins4 (( Ins2 Ins2 S i^i Ins4 SI3 ( Ins2 `'1st i^i (_V X. `'2nd)))"1c) <-> <.{b}, <.{z}, x>.>. e. (( Ins2 Ins2 S i^i Ins4 SI3 ( Ins2 `'1st i^i (_V X. `'2nd)))"1c))
14		elin 3220	. . . . . . . . . . . . 13 |- (<.{a}, <.{b}, <.{z}, x>.>.>. e. ( Ins2 Ins2 S i^i Ins4 SI3 ( Ins2 `'1st i^i (_V X. `'2nd))) <-> (<.{a}, <.{b}, <.{z}, x>.>.>. e. Ins2 Ins2 S /\ <.{a}, <.{b}, <.{z}, x>.>.>. e. Ins4 SI3 ( Ins2 `'1st i^i (_V X. `'2nd))))
15		snex 4112	. . . . . . . . . . . . . . . 16 |- {b} e. _V
16	15	otelins2 5792	. . . . . . . . . . . . . . 15 |- (<.{a}, <.{b}, <.{z}, x>.>.>. e. Ins2 Ins2 S <-> <.{a}, <.{z}, x>.>. e. Ins2 S )
17	5	otelins2 5792	. . . . . . . . . . . . . . 15 |- (<.{a}, <.{z}, x>.>. e. Ins2 S <-> <.{a}, x>. e. S )
18		vex 2863	. . . . . . . . . . . . . . . 16 |- a e. _V
19	18, 7	opelssetsn 4761	. . . . . . . . . . . . . . 15 |- (<.{a}, x>. e. S <-> a e. x)
20	16, 17, 19	3bitri 262	. . . . . . . . . . . . . 14 |- (<.{a}, <.{b}, <.{z}, x>.>.>. e. Ins2 Ins2 S <-> a e. x)
21	7	oqelins4 5795	. . . . . . . . . . . . . . 15 |- (<.{a}, <.{b}, <.{z}, x>.>.>. e. Ins4 SI3 ( Ins2 `'1st i^i (_V X. `'2nd)) <-> <.{a}, <.{b}, {z}>.>. e. SI3 ( Ins2 `'1st i^i (_V X. `'2nd)))
22		vex 2863	. . . . . . . . . . . . . . . 16 |- z e. _V
23	18, 9, 22	otsnelsi3 5806	. . . . . . . . . . . . . . 15 |- (<.{a}, <.{b}, {z}>.>. e. SI3 ( Ins2 `'1st i^i (_V X. `'2nd)) <-> <.a, <.b, z>.>. e. ( Ins2 `'1st i^i (_V X. `'2nd)))
24		elin 3220	. . . . . . . . . . . . . . . 16 |- (<.a, <.b, z>.>. e. ( Ins2 `'1st i^i (_V X. `'2nd)) <-> (<.a, <.b, z>.>. e. Ins2 `'1st /\ <.a, <.b, z>.>. e. (_V X. `'2nd)))
25	9	otelins2 5792	. . . . . . . . . . . . . . . . . 18 |- (<.a, <.b, z>.>. e. Ins2 `'1st <-> <.a, z>. e. `'1st)
26		df-br 4641	. . . . . . . . . . . . . . . . . 18 |- (a`'1stz <-> <.a, z>. e. `'1st)
27		brcnv 4893	. . . . . . . . . . . . . . . . . 18 |- (a`'1stz <-> z1sta)
28	25, 26, 27	3bitr2i 264	. . . . . . . . . . . . . . . . 17 |- (<.a, <.b, z>.>. e. Ins2 `'1st <-> z1sta)
29		opelxp 4812	. . . . . . . . . . . . . . . . . . 19 |- (<.a, <.b, z>.>. e. (_V X. `'2nd) <-> (a e. _V /\ <.b, z>. e. `'2nd))
30	18, 29	mpbiran 884	. . . . . . . . . . . . . . . . . 18 |- (<.a, <.b, z>.>. e. (_V X. `'2nd) <-> <.b, z>. e. `'2nd)
31		df-br 4641	. . . . . . . . . . . . . . . . . 18 |- (b`'2ndz <-> <.b, z>. e. `'2nd)
32		brcnv 4893	. . . . . . . . . . . . . . . . . 18 |- (b`'2ndz <-> z2ndb)
33	30, 31, 32	3bitr2i 264	. . . . . . . . . . . . . . . . 17 |- (<.a, <.b, z>.>. e. (_V X. `'2nd) <-> z2ndb)
34	28, 33	anbi12i 678	. . . . . . . . . . . . . . . 16 |- ((<.a, <.b, z>.>. e. Ins2 `'1st /\ <.a, <.b, z>.>. e. (_V X. `'2nd)) <-> (z1sta /\ z2ndb))
35	18, 9	op1st2nd 5791	. . . . . . . . . . . . . . . 16 |- ((z1sta /\ z2ndb) <-> z = <.a, b>.)
36	24, 34, 35	3bitri 262	. . . . . . . . . . . . . . 15 |- (<.a, <.b, z>.>. e. ( Ins2 `'1st i^i (_V X. `'2nd)) <-> z = <.a, b>.)
37	21, 23, 36	3bitri 262	. . . . . . . . . . . . . 14 |- (<.{a}, <.{b}, <.{z}, x>.>.>. e. Ins4 SI3 ( Ins2 `'1st i^i (_V X. `'2nd)) <-> z = <.a, b>.)
38	20, 37	anbi12i 678	. . . . . . . . . . . . 13 |- ((<.{a}, <.{b}, <.{z}, x>.>.>. e. Ins2 Ins2 S /\ <.{a}, <.{b}, <.{z}, x>.>.>. e. Ins4 SI3 ( Ins2 `'1st i^i (_V X. `'2nd))) <-> (a e. x /\ z = <.a, b>.))
39	14, 38	bitri 240	. . . . . . . . . . . 12 |- (<.{a}, <.{b}, <.{z}, x>.>.>. e. ( Ins2 Ins2 S i^i Ins4 SI3 ( Ins2 `'1st i^i (_V X. `'2nd))) <-> (a e. x /\ z = <.a, b>.))
40	39	exbii 1582	. . . . . . . . . . 11 |- (E.a<.{a}, <.{b}, <.{z}, x>.>.>. e. ( Ins2 Ins2 S i^i Ins4 SI3 ( Ins2 `'1st i^i (_V X. `'2nd))) <-> E.a(a e. x /\ z = <.a, b>.))
41		elima1c 4948	. . . . . . . . . . 11 |- (<.{b}, <.{z}, x>.>. e. (( Ins2 Ins2 S i^i Ins4 SI3 ( Ins2 `'1st i^i (_V X. `'2nd)))"1c) <-> E.a<.{a}, <.{b}, <.{z}, x>.>.>. e. ( Ins2 Ins2 S i^i Ins4 SI3 ( Ins2 `'1st i^i (_V X. `'2nd))))
42		df-rex 2621	. . . . . . . . . . 11 |- (E.a e. x z = <.a, b>. <-> E.a(a e. x /\ z = <.a, b>.))
43	40, 41, 42	3bitr4i 268	. . . . . . . . . 10 |- (<.{b}, <.{z}, x>.>. e. (( Ins2 Ins2 S i^i Ins4 SI3 ( Ins2 `'1st i^i (_V X. `'2nd)))"1c) <-> E.a e. x z = <.a, b>.)
44	13, 43	bitri 240	. . . . . . . . 9 |- (<.{b}, <.{z}, <.x, y>.>.>. e. Ins4 (( Ins2 Ins2 S i^i Ins4 SI3 ( Ins2 `'1st i^i (_V X. `'2nd)))"1c) <-> E.a e. x z = <.a, b>.)
45	12, 44	anbi12i 678	. . . . . . . 8 |- ((<.{b}, <.{z}, <.x, y>.>.>. e. Ins2 Ins2 S /\ <.{b}, <.{z}, <.x, y>.>.>. e. Ins4 (( Ins2 Ins2 S i^i Ins4 SI3 ( Ins2 `'1st i^i (_V X. `'2nd)))"1c)) <-> (b e. y /\ E.a e. x z = <.a, b>.))
46	4, 45	bitri 240	. . . . . . 7 |- (<.{b}, <.{z}, <.x, y>.>.>. e. ( Ins2 Ins2 S i^i Ins4 (( Ins2 Ins2 S i^i Ins4 SI3 ( Ins2 `'1st i^i (_V X. `'2nd)))"1c)) <-> (b e. y /\ E.a e. x z = <.a, b>.))
47	46	exbii 1582	. . . . . 6 |- (E.b<.{b}, <.{z}, <.x, y>.>.>. e. ( Ins2 Ins2 S i^i Ins4 (( Ins2 Ins2 S i^i Ins4 SI3 ( Ins2 `'1st i^i (_V X. `'2nd)))"1c)) <-> E.b(b e. y /\ E.a e. x z = <.a, b>.))
48		elima1c 4948	. . . . . 6 |- (<.{z}, <.x, y>.>. e. (( Ins2 Ins2 S i^i Ins4 (( Ins2 Ins2 S i^i Ins4 SI3 ( Ins2 `'1st i^i (_V X. `'2nd)))"1c))"1c) <-> E.b<.{b}, <.{z}, <.x, y>.>.>. e. ( Ins2 Ins2 S i^i Ins4 (( Ins2 Ins2 S i^i Ins4 SI3 ( Ins2 `'1st i^i (_V X. `'2nd)))"1c)))
49		df-rex 2621	. . . . . 6 |- (E.b e. y E.a e. x z = <.a, b>. <-> E.b(b e. y /\ E.a e. x z = <.a, b>.))
50	47, 48, 49	3bitr4i 268	. . . . 5 |- (<.{z}, <.x, y>.>. e. (( Ins2 Ins2 S i^i Ins4 (( Ins2 Ins2 S i^i Ins4 SI3 ( Ins2 `'1st i^i (_V X. `'2nd)))"1c))"1c) <-> E.b e. y E.a e. x z = <.a, b>.)
51	2, 3, 50	3bitr4ri 269	. . . 4 |- (<.{z}, <.x, y>.>. e. (( Ins2 Ins2 S i^i Ins4 (( Ins2 Ins2 S i^i Ins4 SI3 ( Ins2 `'1st i^i (_V X. `'2nd)))"1c))"1c) <-> z e. (x X. y))
52	51	releqmpt2 5810	. . 3 |- (((_V X. _V) X. _V) \ (( Ins2 S (+) Ins3 (( Ins2 Ins2 S i^i Ins4 (( Ins2 Ins2 S i^i Ins4 SI3 ( Ins2 `'1st i^i (_V X. `'2nd)))"1c))"1c))"1c)) = (x e. _V, y e. _V |-> (x X. y))
53	1, 52	eqtr4i 2376	. 2 |- Cross = (((_V X. _V) X. _V) \ (( Ins2 S (+) Ins3 (( Ins2 Ins2 S i^i Ins4 (( Ins2 Ins2 S i^i Ins4 SI3 ( Ins2 `'1st i^i (_V X. `'2nd)))"1c))"1c))"1c))
54		vvex 4110	. . 3 |- _V e. _V
55		ssetex 4745	. . . . . . 7 |- S e. _V
56	55	ins2ex 5798	. . . . . 6 |- Ins2 S e. _V
57	56	ins2ex 5798	. . . . 5 |- Ins2 Ins2 S e. _V
58		1stex 4740	. . . . . . . . . . . . 13 |- 1st e. _V
59	58	cnvex 5103	. . . . . . . . . . . 12 |- `'1st e. _V
60	59	ins2ex 5798	. . . . . . . . . . 11 |- Ins2 `'1st e. _V
61		2ndex 5113	. . . . . . . . . . . . 13 |- 2nd e. _V
62	61	cnvex 5103	. . . . . . . . . . . 12 |- `'2nd e. _V
63	54, 62	xpex 5116	. . . . . . . . . . 11 |- (_V X. `'2nd) e. _V
64	60, 63	inex 4106	. . . . . . . . . 10 |- ( Ins2 `'1st i^i (_V X. `'2nd)) e. _V
65	64	si3ex 5807	. . . . . . . . 9 |- SI3 ( Ins2 `'1st i^i (_V X. `'2nd)) e. _V
66	65	ins4ex 5800	. . . . . . . 8 |- Ins4 SI3 ( Ins2 `'1st i^i (_V X. `'2nd)) e. _V
67	57, 66	inex 4106	. . . . . . 7 |- ( Ins2 Ins2 S i^i Ins4 SI3 ( Ins2 `'1st i^i (_V X. `'2nd))) e. _V
68		1cex 4143	. . . . . . 7 |- 1c e. _V
69	67, 68	imaex 4748	. . . . . 6 |- (( Ins2 Ins2 S i^i Ins4 SI3 ( Ins2 `'1st i^i (_V X. `'2nd)))"1c) e. _V
70	69	ins4ex 5800	. . . . 5 |- Ins4 (( Ins2 Ins2 S i^i Ins4 SI3 ( Ins2 `'1st i^i (_V X. `'2nd)))"1c) e. _V
71	57, 70	inex 4106	. . . 4 |- ( Ins2 Ins2 S i^i Ins4 (( Ins2 Ins2 S i^i Ins4 SI3 ( Ins2 `'1st i^i (_V X. `'2nd)))"1c)) e. _V
72	71, 68	imaex 4748	. . 3 |- (( Ins2 Ins2 S i^i Ins4 (( Ins2 Ins2 S i^i Ins4 SI3 ( Ins2 `'1st i^i (_V X. `'2nd)))"1c))"1c) e. _V
73	54, 54, 72	mpt2exlem 5812	. 2 |- (((_V X. _V) X. _V) \ (( Ins2 S (+) Ins3 (( Ins2 Ins2 S i^i Ins4 (( Ins2 Ins2 S i^i Ins4 SI3 ( Ins2 `'1st i^i (_V X. `'2nd)))"1c))"1c))"1c)) e. _V
74	53, 73	eqeltri 2423	1 |- Cross e. _V

Syntax hints:   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  E.wrex 2616  _Vcvv 2860   \ cdif 3207   i^i cin 3209   (+) csymdif 3210  {csn 3738  1_cc1c 4135  <.cop 4562   class class class wbr 4640  1stc1st 4718   S csset 4720  "cima 4723   X. cxp 4771  `'ccnv 4772  2ndc2nd 4784   |-> cmpt2 5654   Ins2 cins2 5750   Ins3 cins3 5752   Ins4 cins4 5756   SI₃ csi3 5758   Cross ccross 5764

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-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-xp 4785  df-cnv 4786  df-2nd 4798  df-oprab 5529  df-mpt2 5655  df-txp 5737  df-ins2 5751  df-ins3 5753  df-ins4 5757  df-si3 5759  df-cross 5765

This theorem is referenced by:  ovmuc  6131  mucex  6134