Theorem xp11m 4789

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 4775	. . 3 |- ( ( E. x x e. A /\ E. y y e. B ) <-> E. z z e. ( A X. B ) )
2		anidm 388	. . . . . 6 |- ( ( E. z z e. ( A X. B ) /\ E. z z e. ( A X. B ) ) <-> E. z z e. ( A X. B ) )
3		eleq2 2143	. . . . . . . 8 |- ( ( A X. B ) = ( C X. D ) -> ( z e. ( A X. B ) <-> z e. ( C X. D ) ) )
4	3	exbidv 1747	. . . . . . 7 |- ( ( A X. B ) = ( C X. D ) -> ( E. z z e. ( A X. B ) <-> E. z z e. ( C X. D ) ) )
5	4	anbi2d 452	. . . . . 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 192	. . . . 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 3052	. . . . . . . 8 |- ( ( A X. B ) = ( C X. D ) -> ( A X. B ) C_ ( C X. D ) )
8		ssxpbm 4786	. . . . . . . 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 153	. . . . . . 7 |- ( ( A X. B ) = ( C X. D ) -> ( E. z z e. ( A X. B ) -> ( A C_ C /\ B C_ D ) ) )
10		eqimss2 3053	. . . . . . . 8 |- ( ( A X. B ) = ( C X. D ) -> ( C X. D ) C_ ( A X. B ) )
11		ssxpbm 4786	. . . . . . . 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 153	. . . . . . 7 |- ( ( A X. B ) = ( C X. D ) -> ( E. z z e. ( C X. D ) -> ( C C_ A /\ D C_ B ) ) )
13	9, 12	anim12d 328	. . . . . 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 551	. . . . . . 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 3015	. . . . . . . 8 |- ( A = C <-> ( A C_ C /\ C C_ A ) )
16		eqss 3015	. . . . . . . 8 |- ( B = D <-> ( B C_ D /\ D C_ B ) )
17	15, 16	anbi12i 448	. . . . . . 7 |- ( ( A = C /\ B = D ) <-> ( ( A C_ C /\ C C_ A ) /\ ( B C_ D /\ D C_ B ) ) )
18	14, 17	bitr4i 185	. . . . . 6 |- ( ( ( A C_ C /\ B C_ D ) /\ ( C C_ A /\ D C_ B ) ) <-> ( A = C /\ B = D ) )
19	13, 18	syl6ib 159	. . . . 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 148	. . . 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 119	. 2 |- ( ( E. x x e. A /\ E. y y e. B ) -> ( ( A X. B ) = ( C X. D ) -> ( A = C /\ B = D ) ) )
23		xpeq12 4390	. 2 |- ( ( A = C /\ B = D ) -> ( A X. B ) = ( C X. D ) )
24	22, 23	impbid1 140	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 102   <-> wb 103   = wceq 1285   E.wex 1422   e. wcel 1434   C_ wss 2974   X. cxp 4369

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-xp 4377  df-rel 4378  df-cnv 4379  df-dm 4381  df-rn 4382

This theorem is referenced by: (None)