Theorem ssxp1 5180

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 4958	. . . . . 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 4936	. . . . . 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 3265	. . . 4 |- ( ( E. x x e. C /\ ( A X. C ) C_ ( B X. C ) ) -> A C_ dom ( B X. C ) )
6		dmxpss 5174	. . . 4 |- dom ( B X. C ) C_ B
7	5, 6	sstrdi 3240	. . 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 4842	. 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 1398   E.wex 1541   e. wcel 2202   C_ wss 3201   X. cxp 4729   dom cdm 4731

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-dm 4741

This theorem is referenced by:  xpcan2m  5184