Theorem ssxp1 5173

Description: Cross product subset cancellation. (Contributed by Jim Kingdon, 14-Dec-2018.)

Assertion

Ref	Expression
ssxp1	|- ( E. x x e. C -> ( ( A X. C ) C_ ( B X. C ) <-> A C_ B ) )

Distinct variable group:   x, C

Allowed substitution hints:   A( x)   B( x)

Proof of Theorem ssxp1

Step	Hyp	Ref	Expression
1		dmxpm 4952	. . . . . 6 |- ( E. x x e. C -> dom ( A X. C ) = A )
2	1	adantr 276	. . . . 5 |- ( ( E. x x e. C /\ ( A X. C ) C_ ( B X. C ) ) -> dom ( A X. C ) = A )
3		dmss 4930	. . . . . 6 |- ( ( A X. C ) C_ ( B X. C ) -> dom ( A X. C ) C_ dom ( B X. C ) )
4	3	adantl 277	. . . . 5 |- ( ( E. x x e. C /\ ( A X. C ) C_ ( B X. C ) ) -> dom ( A X. C ) C_ dom ( B X. C ) )
5	2, 4	eqsstrrd 3264	. . . 4 |- ( ( E. x x e. C /\ ( A X. C ) C_ ( B X. C ) ) -> A C_ dom ( B X. C ) )
6		dmxpss 5167	. . . 4 |- dom ( B X. C ) C_ B
7	5, 6	sstrdi 3239	. . 3 |- ( ( E. x x e. C /\ ( A X. C ) C_ ( B X. C ) ) -> A C_ B )
8	7	ex 115	. 2 |- ( E. x x e. C -> ( ( A X. C ) C_ ( B X. C ) -> A C_ B ) )
9		xpss1 4836	. 2 |- ( A C_ B -> ( A X. C ) C_ ( B X. C ) )
10	8, 9	impbid1 142	1 |- ( E. x x e. C -> ( ( A X. C ) C_ ( B X. C ) <-> A C_ B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   E.wex 1540   e. wcel 2202   C_ wss 3200   X. cxp 4723   dom cdm 4725

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-dm 4735

This theorem is referenced by:  xpcan2m  5177