Theorem xpen 6967

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 6858	. . . 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 6858	. . . . 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 6854	. . . . . . 7 |- Rel ~~
8	7	brrelex1i 4736	. . . . . 6 |- ( A ~~ B -> A e. _V )
9	7	brrelex1i 4736	. . . . . 6 |- ( C ~~ D -> C e. _V )
10		xpexg 4807	. . . . . 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 5545	. . . . . . . 8 |- ( f : A -1-1-onto-> B -> f Fn A )
15		dffn5im 5647	. . . . . . . 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 5532	. . . . . . 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 5545	. . . . . . . 8 |- ( g : C -1-1-onto-> D -> g Fn C )
22		dffn5im 5647	. . . . . . . 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 5532	. . . . . . 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 6966	. . . 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 6871	. . . 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 1923	. 2 |- ( ( ( A ~~ B /\ C ~~ D ) /\ f : A -1-1-onto-> B ) -> ( A X. C ) ~~ ( B X. D ) )
31	3, 30	exlimddv 1923	1 |- ( ( A ~~ B /\ C ~~ D ) -> ( A X. C ) ~~ ( B X. D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   E.wex 1516   e. wcel 2178   _Vcvv 2776   <.cop 3646   class class class wbr 4059   |-> cmpt 4121   X. cxp 4691   Fn wfn 5285   -1-1-onto->wf1o 5289   ` cfv 5290   e. cmpo 5969   ~~ cen 6848

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-en 6851

This theorem is referenced by:  xpdjuen  7361  xpnnen  12880  xpomen  12881  qnnen  12917