Theorem djussxp 4499

Description: Disjoint union is a subset of a cross product. (Contributed by Stefan O'Rear, 21-Nov-2014.)

Assertion

Ref	Expression
djussxp	|- U_ x e. A ( { x } X. B ) C_ ( A X. _V )

Distinct variable group:   x, A

Allowed substitution hint:   B( x)

Proof of Theorem djussxp

Step	Hyp	Ref	Expression
1		iunss 3719	. 2 |- ( U_ x e. A ( { x } X. B ) C_ ( A X. _V ) <-> A. x e. A ( { x } X. B ) C_ ( A X. _V ) )
2		snssi 3529	. . 3 |- ( x e. A -> { x } C_ A )
3		ssv 3019	. . 3 |- B C_ _V
4		xpss12 4463	. . 3 |- ( ( { x } C_ A /\ B C_ _V ) -> ( { x } X. B ) C_ ( A X. _V ) )
5	2, 3, 4	sylancl 404	. 2 |- ( x e. A -> ( { x } X. B ) C_ ( A X. _V ) )
6	1, 5	mprgbir 2421	1 |- U_ x e. A ( { x } X. B ) C_ ( A X. _V )

Syntax hints:   e. wcel 1433   _Vcvv 2601   C_ wss 2973   {csn 3398   U_ciun 3678   X. cxp 4361

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-in 2979  df-ss 2986  df-sn 3404  df-iun 3680  df-opab 3840  df-xp 4369

This theorem is referenced by:  djudisj  4770