Theorem xpen 6845

Description: Equinumerosity law for Cartesian product. Proposition 4.22(b) of [Mendelson] p. 254. (Contributed by NM, 24-Jul-2004.)

Assertion

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

Proof of Theorem xpen

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

Step	Hyp	Ref	Expression
1		bren 6747	. . . 4 |- ( A ~~ B <-> E. f f : A -1-1-onto-> B )
2	1	biimpi 120	. . 3 |- ( A ~~ B -> E. f f : A -1-1-onto-> B )
3	2	adantr 276	. 2 |- ( ( A ~~ B /\ C ~~ D ) -> E. f f : A -1-1-onto-> B )
4		bren 6747	. . . . 5 |- ( C ~~ D <-> E. g g : C -1-1-onto-> D )
5	4	biimpi 120	. . . 4 |- ( C ~~ D -> E. g g : C -1-1-onto-> D )
6	5	ad2antlr 489	. . 3 |- ( ( ( A ~~ B /\ C ~~ D ) /\ f : A -1-1-onto-> B ) -> E. g g : C -1-1-onto-> D )
7		relen 6744	. . . . . . 7 |- Rel ~~
8	7	brrelex1i 4670	. . . . . 6 |- ( A ~~ B -> A e. _V )
9	7	brrelex1i 4670	. . . . . 6 |- ( C ~~ D -> C e. _V )
10		xpexg 4741	. . . . . 6 |- ( ( A e. _V /\ C e. _V ) -> ( A X. C ) e. _V )
11	8, 9, 10	syl2an 289	. . . . 5 |- ( ( A ~~ B /\ C ~~ D ) -> ( A X. C ) e. _V )
12	11	ad2antrr 488	. . . 4 |- ( ( ( ( A ~~ B /\ C ~~ D ) /\ f : A -1-1-onto-> B ) /\ g : C -1-1-onto-> D ) -> ( A X. C ) e. _V )
13		simplr 528	. . . . . 6 |- ( ( ( ( A ~~ B /\ C ~~ D ) /\ f : A -1-1-onto-> B ) /\ g : C -1-1-onto-> D ) -> f : A -1-1-onto-> B )
14		f1ofn 5463	. . . . . . . 8 |- ( f : A -1-1-onto-> B -> f Fn A )
15		dffn5im 5562	. . . . . . . 8 |- ( f Fn A -> f = ( x e. A |-> ( f ` x ) ) )
16	14, 15	syl 14	. . . . . . 7 |- ( f : A -1-1-onto-> B -> f = ( x e. A |-> ( f ` x ) ) )
17		f1oeq1 5450	. . . . . . 7 |- ( f = ( x e. A |-> ( f ` x ) ) -> ( f : A -1-1-onto-> B <-> ( x e. A |-> ( f ` x ) ) : A -1-1-onto-> B ) )
18	13, 16, 17	3syl 17	. . . . . 6 |- ( ( ( ( A ~~ B /\ C ~~ D ) /\ f : A -1-1-onto-> B ) /\ g : C -1-1-onto-> D ) -> ( f : A -1-1-onto-> B <-> ( x e. A |-> ( f ` x ) ) : A -1-1-onto-> B ) )
19	13, 18	mpbid 147	. . . . 5 |- ( ( ( ( A ~~ B /\ C ~~ D ) /\ f : A -1-1-onto-> B ) /\ g : C -1-1-onto-> D ) -> ( x e. A |-> ( f ` x ) ) : A -1-1-onto-> B )
20		simpr 110	. . . . . 6 |- ( ( ( ( A ~~ B /\ C ~~ D ) /\ f : A -1-1-onto-> B ) /\ g : C -1-1-onto-> D ) -> g : C -1-1-onto-> D )
21		f1ofn 5463	. . . . . . . 8 |- ( g : C -1-1-onto-> D -> g Fn C )
22		dffn5im 5562	. . . . . . . 8 |- ( g Fn C -> g = ( y e. C |-> ( g ` y ) ) )
23	21, 22	syl 14	. . . . . . 7 |- ( g : C -1-1-onto-> D -> g = ( y e. C |-> ( g ` y ) ) )
24		f1oeq1 5450	. . . . . . 7 |- ( g = ( y e. C |-> ( g ` y ) ) -> ( g : C -1-1-onto-> D <-> ( y e. C |-> ( g ` y ) ) : C -1-1-onto-> D ) )
25	20, 23, 24	3syl 17	. . . . . 6 |- ( ( ( ( A ~~ B /\ C ~~ D ) /\ f : A -1-1-onto-> B ) /\ g : C -1-1-onto-> D ) -> ( g : C -1-1-onto-> D <-> ( y e. C |-> ( g ` y ) ) : C -1-1-onto-> D ) )
26	20, 25	mpbid 147	. . . . 5 |- ( ( ( ( A ~~ B /\ C ~~ D ) /\ f : A -1-1-onto-> B ) /\ g : C -1-1-onto-> D ) -> ( y e. C |-> ( g ` y ) ) : C -1-1-onto-> D )
27	19, 26	xpf1o 6844	. . . 4 |- ( ( ( ( A ~~ B /\ C ~~ D ) /\ f : A -1-1-onto-> B ) /\ g : C -1-1-onto-> D ) -> ( x e. A , y e. C |-> <. ( f ` x ) , ( g ` y ) >. ) : ( A X. C ) -1-1-onto-> ( B X. D ) )
28		f1oeng 6757	. . . 4 |- ( ( ( A X. C ) e. _V /\ ( x e. A , y e. C |-> <. ( f ` x ) , ( g ` y ) >. ) : ( A X. C ) -1-1-onto-> ( B X. D ) ) -> ( A X. C ) ~~ ( B X. D ) )
29	12, 27, 28	syl2anc 411	. . 3 |- ( ( ( ( A ~~ B /\ C ~~ D ) /\ f : A -1-1-onto-> B ) /\ g : C -1-1-onto-> D ) -> ( A X. C ) ~~ ( B X. D ) )
30	6, 29	exlimddv 1898	. 2 |- ( ( ( A ~~ B /\ C ~~ D ) /\ f : A -1-1-onto-> B ) -> ( A X. C ) ~~ ( B X. D ) )
31	3, 30	exlimddv 1898	1 |- ( ( A ~~ B /\ C ~~ D ) -> ( A X. C ) ~~ ( B X. D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   E.wex 1492   e. wcel 2148   _Vcvv 2738   <.cop 3596   class class class wbr 4004   |-> cmpt 4065   X. cxp 4625   Fn wfn 5212   -1-1-onto->wf1o 5216   ` cfv 5217   e. cmpo 5877   ~~ cen 6738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-en 6741

This theorem is referenced by:  xpdjuen  7217  xpnnen  12395  xpomen  12396  qnnen  12432