Theorem ssxp2 5107

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 5099	. . . . . 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 4896	. . . . . 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 3220	. . . 4 |- ( ( E. x x e. C /\ ( C X. A ) C_ ( C X. B ) ) -> A C_ ran ( C X. B ) )
6		rnxpss 5101	. . . 4 |- ran ( C X. B ) C_ B
7	5, 6	sstrdi 3195	. . 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 4774	. 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 1364   E.wex 1506   e. wcel 2167   C_ wss 3157   X. cxp 4661   ran crn 4664

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-xp 4669  df-rel 4670  df-cnv 4671  df-dm 4673  df-rn 4674

This theorem is referenced by:  xpcanm  5109