Theorem xp11m 4759

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 4745	. . 3 |- ( ( E. x x e. A /\ E. y y e. B ) <-> E. z z e. ( A X. B ) )
2		anidm 376	. . . . . 6 |- ( ( E. z z e. ( A X. B ) /\ E. z z e. ( A X. B ) ) <-> E. z z e. ( A X. B ) )
3		eleq2 2101	. . . . . . . 8 |- ( ( A X. B ) = ( C X. D ) -> ( z e. ( A X. B ) <-> z e. ( C X. D ) ) )
4	3	exbidv 1706	. . . . . . 7 |- ( ( A X. B ) = ( C X. D ) -> ( E. z z e. ( A X. B ) <-> E. z z e. ( C X. D ) ) )
5	4	anbi2d 437	. . . . . 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	syl5bbr 183	. . . . 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 2997	. . . . . . . 8 |- ( ( A X. B ) = ( C X. D ) -> ( A X. B ) C_ ( C X. D ) )
8		ssxpbm 4756	. . . . . . . 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 144	. . . . . . 7 |- ( ( A X. B ) = ( C X. D ) -> ( E. z z e. ( A X. B ) -> ( A C_ C /\ B C_ D ) ) )
10		eqimss2 2998	. . . . . . . 8 |- ( ( A X. B ) = ( C X. D ) -> ( C X. D ) C_ ( A X. B ) )
11		ssxpbm 4756	. . . . . . . 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 144	. . . . . . 7 |- ( ( A X. B ) = ( C X. D ) -> ( E. z z e. ( C X. D ) -> ( C C_ A /\ D C_ B ) ) )
13	9, 12	anim12d 318	. . . . . 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 520	. . . . . . 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 2960	. . . . . . . 8 |- ( A = C <-> ( A C_ C /\ C C_ A ) )
16		eqss 2960	. . . . . . . 8 |- ( B = D <-> ( B C_ D /\ D C_ B ) )
17	15, 16	anbi12i 433	. . . . . . 7 |- ( ( A = C /\ B = D ) <-> ( ( A C_ C /\ C C_ A ) /\ ( B C_ D /\ D C_ B ) ) )
18	14, 17	bitr4i 176	. . . . . 6 |- ( ( ( A C_ C /\ B C_ D ) /\ ( C C_ A /\ D C_ B ) ) <-> ( A = C /\ B = D ) )
19	13, 18	syl6ib 150	. . . . 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 139	. . . 4 |- ( ( A X. B ) = ( C X. D ) -> ( E. z z e. ( A X. B ) -> ( A = C /\ B = D ) ) )
21	20	com12 27	. . 3 |- ( E. z z e. ( A X. B ) -> ( ( A X. B ) = ( C X. D ) -> ( A = C /\ B = D ) ) )
22	1, 21	sylbi 114	. 2 |- ( ( E. x x e. A /\ E. y y e. B ) -> ( ( A X. B ) = ( C X. D ) -> ( A = C /\ B = D ) ) )
23		xpeq12 4364	. 2 |- ( ( A = C /\ B = D ) -> ( A X. B ) = ( C X. D ) )
24	22, 23	impbid1 130	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 97   <-> wb 98   = wceq 1243   E.wex 1381   e. wcel 1393   C_ wss 2917   X. cxp 4343

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356

This theorem is referenced by: (None)