Theorem ssxp2 4984

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

Assertion

Ref	Expression
ssxp2	|- ( E. x x e. C -> ( ( C X. A ) C_ ( C X. B ) <-> A C_ B ) )

Distinct variable group:   x, C

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

Proof of Theorem ssxp2

Step	Hyp	Ref	Expression
1		rnxpm 4976	. . . . . 6 |- ( E. x x e. C -> ran ( C X. A ) = A )
2	1	adantr 274	. . . . 5 |- ( ( E. x x e. C /\ ( C X. A ) C_ ( C X. B ) ) -> ran ( C X. A ) = A )
3		rnss 4777	. . . . . 6 |- ( ( C X. A ) C_ ( C X. B ) -> ran ( C X. A ) C_ ran ( C X. B ) )
4	3	adantl 275	. . . . 5 |- ( ( E. x x e. C /\ ( C X. A ) C_ ( C X. B ) ) -> ran ( C X. A ) C_ ran ( C X. B ) )
5	2, 4	eqsstrrd 3139	. . . 4 |- ( ( E. x x e. C /\ ( C X. A ) C_ ( C X. B ) ) -> A C_ ran ( C X. B ) )
6		rnxpss 4978	. . . 4 |- ran ( C X. B ) C_ B
7	5, 6	sstrdi 3114	. . 3 |- ( ( E. x x e. C /\ ( C X. A ) C_ ( C X. B ) ) -> A C_ B )
8	7	ex 114	. 2 |- ( E. x x e. C -> ( ( C X. A ) C_ ( C X. B ) -> A C_ B ) )
9		xpss2 4658	. 2 |- ( A C_ B -> ( C X. A ) C_ ( C X. B ) )
10	8, 9	impbid1 141	1 |- ( E. x x e. C -> ( ( C X. A ) C_ ( C X. B ) <-> A C_ B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   E.wex 1469   e. wcel 1481   C_ wss 3076   X. cxp 4545   ran crn 4548

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-cnv 4555  df-dm 4557  df-rn 4558

This theorem is referenced by:  xpcanm  4986