Theorem xpen 7026

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 6912	. . . 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 6912	. . . . 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 6908	. . . . . . 7 |- Rel ~~
8	7	brrelex1i 4767	. . . . . 6 |- ( A ~~ B -> A e. _V )
9	7	brrelex1i 4767	. . . . . 6 |- ( C ~~ D -> C e. _V )
10		xpexg 4838	. . . . . 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 5581	. . . . . . . 8 |- ( f : A -1-1-onto-> B -> f Fn A )
15		dffn5im 5687	. . . . . . . 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 5568	. . . . . . 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 5581	. . . . . . . 8 |- ( g : C -1-1-onto-> D -> g Fn C )
22		dffn5im 5687	. . . . . . . 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 5568	. . . . . . 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 7025	. . . 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 6925	. . . 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 1945	. 2 |- ( ( ( A ~~ B /\ C ~~ D ) /\ f : A -1-1-onto-> B ) -> ( A X. C ) ~~ ( B X. D ) )
31	3, 30	exlimddv 1945	1 |- ( ( A ~~ B /\ C ~~ D ) -> ( A X. C ) ~~ ( B X. D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   E.wex 1538   e. wcel 2200   _Vcvv 2800   <.cop 3670   class class class wbr 4086   |-> cmpt 4148   X. cxp 4721   Fn wfn 5319   -1-1-onto->wf1o 5323   ` cfv 5324   e. cmpo 6015   ~~ cen 6902

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-en 6905

This theorem is referenced by:  xpdjuen  7423  xpnnen  13005  xpomen  13006  qnnen  13042