Theorem ssxp2 5119

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 5111	. . . . . 6 |- ( E. x x e. C -> ran ( C X. A ) = A )
2	1	adantr 276	. . . . 5 |- ( ( E. x x e. C /\ ( C X. A ) C_ ( C X. B ) ) -> ran ( C X. A ) = A )
3		rnss 4907	. . . . . 6 |- ( ( C X. A ) C_ ( C X. B ) -> ran ( C X. A ) C_ ran ( C X. B ) )
4	3	adantl 277	. . . . 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 3229	. . . 4 |- ( ( E. x x e. C /\ ( C X. A ) C_ ( C X. B ) ) -> A C_ ran ( C X. B ) )
6		rnxpss 5113	. . . 4 |- ran ( C X. B ) C_ B
7	5, 6	sstrdi 3204	. . 3 |- ( ( E. x x e. C /\ ( C X. A ) C_ ( C X. B ) ) -> A C_ B )
8	7	ex 115	. 2 |- ( E. x x e. C -> ( ( C X. A ) C_ ( C X. B ) -> A C_ B ) )
9		xpss2 4785	. 2 |- ( A C_ B -> ( C X. A ) C_ ( C X. B ) )
10	8, 9	impbid1 142	1 |- ( E. x x e. C -> ( ( C X. A ) C_ ( C X. B ) <-> A C_ B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1372   E.wex 1514   e. wcel 2175   C_ wss 3165   X. cxp 4672   ran crn 4675

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-xp 4680  df-rel 4681  df-cnv 4682  df-dm 4684  df-rn 4685

This theorem is referenced by:  xpcanm  5121