Theorem djussxp 4684

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 3854	. 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 3664	. . 3 |- ( x e. A -> { x } C_ A )
3		ssv 3119	. . 3 |- B C_ _V
4		xpss12 4646	. . 3 |- ( ( { x } C_ A /\ B C_ _V ) -> ( { x } X. B ) C_ ( A X. _V ) )
5	2, 3, 4	sylancl 409	. 2 |- ( x e. A -> ( { x } X. B ) C_ ( A X. _V ) )
6	1, 5	mprgbir 2490	1 |- U_ x e. A ( { x } X. B ) C_ ( A X. _V )

Syntax hints:   e. wcel 1480   _Vcvv 2686   C_ wss 3071   {csn 3527   U_ciun 3813   X. cxp 4537

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-sn 3533  df-iun 3815  df-opab 3990  df-xp 4545

This theorem is referenced by:  djudisj  4966