Theorem djussxp 5267

Description: Disjoint union is a subset of a Cartesian 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 4561	. 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 4339	. . 3 |- ( x e. A -> { x } C_ A )
3		ssv 3625	. . 3 |- B C_ _V
4		xpss12 5225	. . 3 |- ( ( { x } C_ A /\ B C_ _V ) -> ( { x } X. B ) C_ ( A X. _V ) )
5	2, 3, 4	sylancl 694	. 2 |- ( x e. A -> ( { x } X. B ) C_ ( A X. _V ) )
6	1, 5	mprgbir 2927	1 |- U_ x e. A ( { x } X. B ) C_ ( A X. _V )

Syntax hints:   e. wcel 1990   _Vcvv 3200   C_ wss 3574   {csn 4177   U_ciun 4520   X. cxp 5112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-in 3581  df-ss 3588  df-sn 4178  df-iun 4522  df-opab 4713  df-xp 5120

This theorem is referenced by:  djudisj  5561  iundom2g  9362