Theorem xpexg 5115

Description: The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. (Contributed by set.mm contributors, 14-Aug-1994.)

Assertion

Ref	Expression
xpexg	|- ((A e. V /\ B e. W) -> (A X. B) e. _V)

Proof of Theorem xpexg

Step	Hyp	Ref	Expression
1		dfxp2 5114	. 2 |- (A X. B) = ((`'1st"A) i^i (`'2nd"B))
2		1stex 4740	. . . . 5 |- 1st e. _V
3	2	cnvex 5103	. . . 4 |- `'1st e. _V
4		imaexg 4747	. . . 4 |- ((`'1st e. _V /\ A e. V) -> (`'1st"A) e. _V)
5	3, 4	mpan 651	. . 3 |- (A e. V -> (`'1st"A) e. _V)
6		2ndex 5113	. . . . 5 |- 2nd e. _V
7	6	cnvex 5103	. . . 4 |- `'2nd e. _V
8		imaexg 4747	. . . 4 |- ((`'2nd e. _V /\ B e. W) -> (`'2nd"B) e. _V)
9	7, 8	mpan 651	. . 3 |- (B e. W -> (`'2nd"B) e. _V)
10		inexg 4101	. . 3 |- (((`'1st"A) e. _V /\ (`'2nd"B) e. _V) -> ((`'1st"A) i^i (`'2nd"B)) e. _V)
11	5, 9, 10	syl2an 463	. 2 |- ((A e. V /\ B e. W) -> ((`'1st"A) i^i (`'2nd"B)) e. _V)
12	1, 11	syl5eqel 2437	1 |- ((A e. V /\ B e. W) -> (A X. B) e. _V)

Syntax hints:   -> wi 4   /\ wa 358   e. wcel 1710  _Vcvv 2860   i^i cin 3209  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-swap 4725  df-co 4727  df-ima 4728  df-xp 4785  df-cnv 4786  df-2nd 4798

This theorem is referenced by:  xpex  5116  resexg  5117  ovcross  5846  fullfunexg  5860  pmex  6006  ovcelem1  6172  fnfreclem1  6318