Theorem ssxp1 4931

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 4717	. . . . . 6 |- ( E. x x e. C -> dom ( A X. C ) = A )
2	1	adantr 272	. . . . 5 |- ( ( E. x x e. C /\ ( A X. C ) C_ ( B X. C ) ) -> dom ( A X. C ) = A )
3		dmss 4696	. . . . . 6 |- ( ( A X. C ) C_ ( B X. C ) -> dom ( A X. C ) C_ dom ( B X. C ) )
4	3	adantl 273	. . . . 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 3098	. . . 4 |- ( ( E. x x e. C /\ ( A X. C ) C_ ( B X. C ) ) -> A C_ dom ( B X. C ) )
6		dmxpss 4925	. . . 4 |- dom ( B X. C ) C_ B
7	5, 6	syl6ss 3073	. . 3 |- ( ( E. x x e. C /\ ( A X. C ) C_ ( B X. C ) ) -> A C_ B )
8	7	ex 114	. 2 |- ( E. x x e. C -> ( ( A X. C ) C_ ( B X. C ) -> A C_ B ) )
9		xpss1 4607	. 2 |- ( A C_ B -> ( A X. C ) C_ ( B X. C ) )
10	8, 9	impbid1 141	1 |- ( E. x x e. C -> ( ( A X. C ) C_ ( B X. C ) <-> A C_ B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1312   E.wex 1449   e. wcel 1461   C_ wss 3035   X. cxp 4495   dom cdm 4497

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-br 3894  df-opab 3948  df-xp 4503  df-dm 4507

This theorem is referenced by:  xpcan2m  4935