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 ∈ V ∧ B ∈ W) → (A × B) ∈ V)

Proof of Theorem xpexg

Step	Hyp	Ref	Expression
1		dfxp2 5114	. 2 ⊢ (A × B) = ((◡1st “ A) ∩ (◡2nd “ B))
2		1stex 4740	. . . . 5 ⊢ 1st ∈ V
3	2	cnvex 5103	. . . 4 ⊢ ◡1st ∈ V
4		imaexg 4747	. . . 4 ⊢ ((◡1st ∈ V ∧ A ∈ V) → (◡1st “ A) ∈ V)
5	3, 4	mpan 651	. . 3 ⊢ (A ∈ V → (◡1st “ A) ∈ V)
6		2ndex 5113	. . . . 5 ⊢ 2nd ∈ V
7	6	cnvex 5103	. . . 4 ⊢ ◡2nd ∈ V
8		imaexg 4747	. . . 4 ⊢ ((◡2nd ∈ V ∧ B ∈ W) → (◡2nd “ B) ∈ V)
9	7, 8	mpan 651	. . 3 ⊢ (B ∈ W → (◡2nd “ B) ∈ V)
10		inexg 4101	. . 3 ⊢ (((◡1st “ A) ∈ V ∧ (◡2nd “ B) ∈ V) → ((◡1st “ A) ∩ (◡2nd “ B)) ∈ V)
11	5, 9, 10	syl2an 463	. 2 ⊢ ((A ∈ V ∧ B ∈ W) → ((◡1st “ A) ∩ (◡2nd “ B)) ∈ V)
12	1, 11	syl5eqel 2437	1 ⊢ ((A ∈ V ∧ B ∈ W) → (A × B) ∈ V)

Syntax hints:   → wi 4   ∧ wa 358   ∈ wcel 1710  Vcvv 2860   ∩ cin 3209  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-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