Theorem ssxp2 5041

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 5033	. . . . . 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 4834	. . . . . 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 3179	. . . 4 |- ( ( E. x x e. C /\ ( C X. A ) C_ ( C X. B ) ) -> A C_ ran ( C X. B ) )
6		rnxpss 5035	. . . 4 |- ran ( C X. B ) C_ B
7	5, 6	sstrdi 3154	. . 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 4715	. 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 1343   E.wex 1480   e. wcel 2136   C_ wss 3116   X. cxp 4602   ran crn 4605

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-dm 4614  df-rn 4615

This theorem is referenced by:  xpcanm  5043