Theorem uniixp 6889

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 6888	. . . . 5 |- ( f e. X_ x e. A B -> f : A --> U_ x e. A B )
2		fssxp 5502	. . . . 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 3659	. . . 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 3231	. 2 |- X_ x e. A B C_ ~P ( A X. U_ x e. A B )
7		sspwuni 4055	. 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 3200   ~Pcpw 3652   U.cuni 3893   U_ciun 3970   X. cxp 4723   -->wf 5322   X_cixp 6866

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-ixp 6867

This theorem is referenced by:  ixpexgg  6890