Theorem ssxp1 5047

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 4831	. . . . . 6 |- ( E. x x e. C -> dom ( A X. C ) = A )
2	1	adantr 274	. . . . 5 |- ( ( E. x x e. C /\ ( A X. C ) C_ ( B X. C ) ) -> dom ( A X. C ) = A )
3		dmss 4810	. . . . . 6 |- ( ( A X. C ) C_ ( B X. C ) -> dom ( A X. C ) C_ dom ( B X. C ) )
4	3	adantl 275	. . . . 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 3184	. . . 4 |- ( ( E. x x e. C /\ ( A X. C ) C_ ( B X. C ) ) -> A C_ dom ( B X. C ) )
6		dmxpss 5041	. . . 4 |- dom ( B X. C ) C_ B
7	5, 6	sstrdi 3159	. . 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 4721	. 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 1348   E.wex 1485   e. wcel 2141   C_ wss 3121   X. cxp 4609   dom cdm 4611

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-dm 4621

This theorem is referenced by:  xpcan2m  5051