Theorem xpen 7098

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 6983	. . . 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 6983	. . . . 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 6979	. . . . . . 7 |- Rel ~~
8	7	brrelex1i 4793	. . . . . 6 |- ( A ~~ B -> A e. _V )
9	7	brrelex1i 4793	. . . . . 6 |- ( C ~~ D -> C e. _V )
10		xpexg 4864	. . . . . 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 529	. . . . . 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 5615	. . . . . . . 8 |- ( f : A -1-1-onto-> B -> f Fn A )
15		dffn5im 5722	. . . . . . . 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 5602	. . . . . . 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 5615	. . . . . . . 8 |- ( g : C -1-1-onto-> D -> g Fn C )
22		dffn5im 5722	. . . . . . . 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 5602	. . . . . . 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 7097	. . . 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 6996	. . . 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 1948	. 2 |- ( ( ( A ~~ B /\ C ~~ D ) /\ f : A -1-1-onto-> B ) -> ( A X. C ) ~~ ( B X. D ) )
31	3, 30	exlimddv 1948	1 |- ( ( A ~~ B /\ C ~~ D ) -> ( A X. C ) ~~ ( B X. D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2203   _Vcvv 2813   <.cop 3692   class class class wbr 4109   |-> cmpt 4171   X. cxp 4747   Fn wfn 5347   -1-1-onto->wf1o 5351   ` cfv 5352   e. cmpo 6052   ~~ cen 6973

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-en 6976

This theorem is referenced by:  xpdjuen  7525  xpnnen  13145  xpomen  13146  qnnen  13182