Theorem djussxp 4841

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 3982	. 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 3788	. . 3 |- ( x e. A -> { x } C_ A )
3		ssv 3223	. . 3 |- B C_ _V
4		xpss12 4800	. . 3 |- ( ( { x } C_ A /\ B C_ _V ) -> ( { x } X. B ) C_ ( A X. _V ) )
5	2, 3, 4	sylancl 413	. 2 |- ( x e. A -> ( { x } X. B ) C_ ( A X. _V ) )
6	1, 5	mprgbir 2566	1 |- U_ x e. A ( { x } X. B ) C_ ( A X. _V )

Syntax hints:   e. wcel 2178   _Vcvv 2776   C_ wss 3174   {csn 3643   U_ciun 3941   X. cxp 4691

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-in 3180  df-ss 3187  df-sn 3649  df-iun 3943  df-opab 4122  df-xp 4699

This theorem is referenced by:  djudisj  5129