Theorem xp11m 5109

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 5092	. . 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 2260	. . . . . . . 8 |- ( ( A X. B ) = ( C X. D ) -> ( z e. ( A X. B ) <-> z e. ( C X. D ) ) )
4	3	exbidv 1839	. . . . . . 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 3238	. . . . . . . 8 |- ( ( A X. B ) = ( C X. D ) -> ( A X. B ) C_ ( C X. D ) )
8		ssxpbm 5106	. . . . . . . 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 3239	. . . . . . . 8 |- ( ( A X. B ) = ( C X. D ) -> ( C X. D ) C_ ( A X. B ) )
11		ssxpbm 5106	. . . . . . . 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 586	. . . . . . 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 3199	. . . . . . . 8 |- ( A = C <-> ( A C_ C /\ C C_ A ) )
16		eqss 3199	. . . . . . . 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 4683	. 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 1364   E.wex 1506   e. wcel 2167   C_ wss 3157   X. cxp 4662

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-xp 4670  df-rel 4671  df-cnv 4672  df-dm 4674  df-rn 4675

This theorem is referenced by:  cc2lem  7349  lmodfopnelem1  13956