Theorem xpen 7149

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 7034	. . . 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 7034	. . . . 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 493	. . 3 |- ( ( ( A ~~ B /\ C ~~ D ) /\ f : A -1-1-onto-> B ) -> E. g g : C -1-1-onto-> D )
7		relen 7030	. . . . . . 7 |- Rel ~~
8	7	brrelex1i 4820	. . . . . 6 |- ( A ~~ B -> A e. _V )
9	7	brrelex1i 4820	. . . . . 6 |- ( C ~~ D -> C e. _V )
10		xpexg 4891	. . . . . 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 492	. . . 4 |- ( ( ( ( A ~~ B /\ C ~~ D ) /\ f : A -1-1-onto-> B ) /\ g : C -1-1-onto-> D ) -> ( A X. C ) e. _V )
13		simplr 533	. . . . . 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 5643	. . . . . . . 8 |- ( f : A -1-1-onto-> B -> f Fn A )
15		dffn5im 5751	. . . . . . . 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 5630	. . . . . . 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 5643	. . . . . . . 8 |- ( g : C -1-1-onto-> D -> g Fn C )
22		dffn5im 5751	. . . . . . . 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 5630	. . . . . . 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 7148	. . . 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 7047	. . . 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 415	. . 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 1954	. 2 |- ( ( ( A ~~ B /\ C ~~ D ) /\ f : A -1-1-onto-> B ) -> ( A X. C ) ~~ ( B X. D ) )
31	3, 30	exlimddv 1954	1 |- ( ( A ~~ B /\ C ~~ D ) -> ( A X. C ) ~~ ( B X. D ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   E.wex 1545   e. wcel 2209   _Vcvv 2821   <.cop 3712   class class class wbr 4131   |-> cmpt 4193   X. cxp 4774   Fn wfn 5375   -1-1-onto->wf1o 5379   ` cfv 5380   e. cmpo 6090   ~~ cen 7024

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-en 7027

This theorem is used by:  xpdjuen  7580  xpnnen  13343  xpomen  13344  qnnen  13380