Theorem xpen 6056

Description: Equinumerosity law for cross product. Proposition 4.22(b) of [Mendelson] p. 254. (Contributed by set.mm contributors, 24-Jul-2004.) (Revised by set.mm contributors, 9-Mar-2013.)

Assertion

Ref	Expression
xpen	|- ((A ~~ B /\ C ~~ D) -> (A X. C) ~~ (B X. D))

Proof of Theorem xpen

Dummy variables f g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bren 6031	. . . 4 |- (A ~~ B <-> E.f f:A-1-1-onto->B)
2		bren 6031	. . . 4 |- (C ~~ D <-> E.g g:C-1-1-onto->D)
3	1, 2	anbi12i 678	. . 3 |- ((A ~~ B /\ C ~~ D) <-> (E.f f:A-1-1-onto->B /\ E.g g:C-1-1-onto->D))
4		eeanv 1913	. . 3 |- (E.fE.g(f:A-1-1-onto->B /\ g:C-1-1-onto->D) <-> (E.f f:A-1-1-onto->B /\ E.g g:C-1-1-onto->D))
5	3, 4	bitr4i 243	. 2 |- ((A ~~ B /\ C ~~ D) <-> E.fE.g(f:A-1-1-onto->B /\ g:C-1-1-onto->D))
6		f1opprod 5845	. . . 4 |- ((f:A-1-1-onto->B /\ g:C-1-1-onto->D) -> PProd (f, g):(A X. C)-1-1-onto->(B X. D))
7		vex 2863	. . . . . 6 |- f e. _V
8		vex 2863	. . . . . 6 |- g e. _V
9	7, 8	pprodex 5839	. . . . 5 |- PProd (f, g) e. _V
10	9	f1oen 6034	. . . 4 |- ( PProd (f, g):(A X. C)-1-1-onto->(B X. D) -> (A X. C) ~~ (B X. D))
11	6, 10	syl 15	. . 3 |- ((f:A-1-1-onto->B /\ g:C-1-1-onto->D) -> (A X. C) ~~ (B X. D))
12	11	exlimivv 1635	. 2 |- (E.fE.g(f:A-1-1-onto->B /\ g:C-1-1-onto->D) -> (A X. C) ~~ (B X. D))
13	5, 12	sylbi 187	1 |- ((A ~~ B /\ C ~~ D) -> (A X. C) ~~ (B X. D))

Syntax hints:   -> wi 4   /\ wa 358  E.wex 1541   class class class wbr 4640   X. cxp 4771  -1-1-onto->wf1o 4781   PProd cpprod 5738   ~~ cen 6029

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-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-2nd 4798  df-txp 5737  df-pprod 5739  df-en 6030

This theorem is referenced by:  mucnc  6132