Theorem xp11m 5182

Description: The cross product of inhabited classes is one-to-one. (Contributed by Jim Kingdon, 13-Dec-2018.)

Assertion

Ref	Expression
xp11m	|- ( ( E. x x e. A /\ E. y y e. B ) -> ( ( A X. B ) = ( C X. D ) <-> ( A = C /\ B = D ) ) )

Distinct variable groups:   x, A   y, B

Allowed substitution hints:   A( y)   B( x)   C( x, y)   D( x, y)

Proof of Theorem xp11m

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpm 5165	. . 3 |- ( ( E. x x e. A /\ E. y y e. B ) <-> E. z z e. ( A X. B ) )
2		anidm 396	. . . . . 6 |- ( ( E. z z e. ( A X. B ) /\ E. z z e. ( A X. B ) ) <-> E. z z e. ( A X. B ) )
3		eleq2 2295	. . . . . . . 8 |- ( ( A X. B ) = ( C X. D ) -> ( z e. ( A X. B ) <-> z e. ( C X. D ) ) )
4	3	exbidv 1873	. . . . . . 7 |- ( ( A X. B ) = ( C X. D ) -> ( E. z z e. ( A X. B ) <-> E. z z e. ( C X. D ) ) )
5	4	anbi2d 464	. . . . . 6 |- ( ( A X. B ) = ( C X. D ) -> ( ( E. z z e. ( A X. B ) /\ E. z z e. ( A X. B ) ) <-> ( E. z z e. ( A X. B ) /\ E. z z e. ( C X. D ) ) ) )
6	2, 5	bitr3id 194	. . . . 5 |- ( ( A X. B ) = ( C X. D ) -> ( E. z z e. ( A X. B ) <-> ( E. z z e. ( A X. B ) /\ E. z z e. ( C X. D ) ) ) )
7		eqimss 3282	. . . . . . . 8 |- ( ( A X. B ) = ( C X. D ) -> ( A X. B ) C_ ( C X. D ) )
8		ssxpbm 5179	. . . . . . . 8 |- ( E. z z e. ( A X. B ) -> ( ( A X. B ) C_ ( C X. D ) <-> ( A C_ C /\ B C_ D ) ) )
9	7, 8	syl5ibcom 155	. . . . . . 7 |- ( ( A X. B ) = ( C X. D ) -> ( E. z z e. ( A X. B ) -> ( A C_ C /\ B C_ D ) ) )
10		eqimss2 3283	. . . . . . . 8 |- ( ( A X. B ) = ( C X. D ) -> ( C X. D ) C_ ( A X. B ) )
11		ssxpbm 5179	. . . . . . . 8 |- ( E. z z e. ( C X. D ) -> ( ( C X. D ) C_ ( A X. B ) <-> ( C C_ A /\ D C_ B ) ) )
12	10, 11	syl5ibcom 155	. . . . . . 7 |- ( ( A X. B ) = ( C X. D ) -> ( E. z z e. ( C X. D ) -> ( C C_ A /\ D C_ B ) ) )
13	9, 12	anim12d 335	. . . . . 6 |- ( ( A X. B ) = ( C X. D ) -> ( ( E. z z e. ( A X. B ) /\ E. z z e. ( C X. D ) ) -> ( ( A C_ C /\ B C_ D ) /\ ( C C_ A /\ D C_ B ) ) ) )
14		an4 588	. . . . . . 7 |- ( ( ( A C_ C /\ B C_ D ) /\ ( C C_ A /\ D C_ B ) ) <-> ( ( A C_ C /\ C C_ A ) /\ ( B C_ D /\ D C_ B ) ) )
15		eqss 3243	. . . . . . . 8 |- ( A = C <-> ( A C_ C /\ C C_ A ) )
16		eqss 3243	. . . . . . . 8 |- ( B = D <-> ( B C_ D /\ D C_ B ) )
17	15, 16	anbi12i 460	. . . . . . 7 |- ( ( A = C /\ B = D ) <-> ( ( A C_ C /\ C C_ A ) /\ ( B C_ D /\ D C_ B ) ) )
18	14, 17	bitr4i 187	. . . . . 6 |- ( ( ( A C_ C /\ B C_ D ) /\ ( C C_ A /\ D C_ B ) ) <-> ( A = C /\ B = D ) )
19	13, 18	imbitrdi 161	. . . . 5 |- ( ( A X. B ) = ( C X. D ) -> ( ( E. z z e. ( A X. B ) /\ E. z z e. ( C X. D ) ) -> ( A = C /\ B = D ) ) )
20	6, 19	sylbid 150	. . . 4 |- ( ( A X. B ) = ( C X. D ) -> ( E. z z e. ( A X. B ) -> ( A = C /\ B = D ) ) )
21	20	com12 30	. . 3 |- ( E. z z e. ( A X. B ) -> ( ( A X. B ) = ( C X. D ) -> ( A = C /\ B = D ) ) )
22	1, 21	sylbi 121	. 2 |- ( ( E. x x e. A /\ E. y y e. B ) -> ( ( A X. B ) = ( C X. D ) -> ( A = C /\ B = D ) ) )
23		xpeq12 4750	. 2 |- ( ( A = C /\ B = D ) -> ( A X. B ) = ( C X. D ) )
24	22, 23	impbid1 142	1 |- ( ( E. x x e. A /\ E. y y e. B ) -> ( ( A X. B ) = ( C X. D ) <-> ( A = C /\ B = D ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2202   C_ wss 3201   X. cxp 4729

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-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-rel 4738  df-cnv 4739  df-dm 4741  df-rn 4742

This theorem is referenced by:  cc2lem  7545  lmodfopnelem1  14420