Theorem ssxp2 5068

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 5060	. . . . . 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 4859	. . . . . 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 3194	. . . 4 |- ( ( E. x x e. C /\ ( C X. A ) C_ ( C X. B ) ) -> A C_ ran ( C X. B ) )
6		rnxpss 5062	. . . 4 |- ran ( C X. B ) C_ B
7	5, 6	sstrdi 3169	. . 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 4739	. 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 1353   E.wex 1492   e. wcel 2148   C_ wss 3131   X. cxp 4626   ran crn 4629

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-rel 4635  df-cnv 4636  df-dm 4638  df-rn 4639

This theorem is referenced by:  xpcanm  5070