Theorem uniixp 6933

Description: The union of an infinite Cartesian product is included in a Cartesian product. (Contributed by NM, 28-Sep-2006.) (Revised by Mario Carneiro, 24-Jun-2015.)

Assertion

Ref	Expression
uniixp	|- U. X_ x e. A B C_ ( A X. U_ x e. A B )

Distinct variable group:   x, A

Allowed substitution hint:   B( x)

Proof of Theorem uniixp

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ixpf 6932	. . . . 5 |- ( f e. X_ x e. A B -> f : A --> U_ x e. A B )
2		fssxp 5510	. . . . 5 |- ( f : A --> U_ x e. A B -> f C_ ( A X. U_ x e. A B ) )
3	1, 2	syl 14	. . . 4 |- ( f e. X_ x e. A B -> f C_ ( A X. U_ x e. A B ) )
4		velpw 3663	. . . 4 |- ( f e. ~P ( A X. U_ x e. A B ) <-> f C_ ( A X. U_ x e. A B ) )
5	3, 4	sylibr 134	. . 3 |- ( f e. X_ x e. A B -> f e. ~P ( A X. U_ x e. A B ) )
6	5	ssriv 3232	. 2 |- X_ x e. A B C_ ~P ( A X. U_ x e. A B )
7		sspwuni 4060	. 2 |- ( X_ x e. A B C_ ~P ( A X. U_ x e. A B ) <-> U. X_ x e. A B C_ ( A X. U_ x e. A B ) )
8	6, 7	mpbi 145	1 |- U. X_ x e. A B C_ ( A X. U_ x e. A B )

Syntax hints:   e. wcel 2202   C_ wss 3201   ~Pcpw 3656   U.cuni 3898   U_ciun 3975   X. cxp 4729   -->wf 5329   X_cixp 6910

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-ixp 6911

This theorem is referenced by:  ixpexgg  6934