Theorem uniixp 6993

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 6992	. . . . 5 |- ( f e. X_ x e. A B -> f : A --> U_ x e. A B )
2		fssxp 5550	. . . . 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 3692	. . . 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 3252	. 2 |- X_ x e. A B C_ ~P ( A X. U_ x e. A B )
7		sspwuni 4092	. 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 2209   C_ wss 3220   ~Pcpw 3685   U.cuni 3930   U_ciun 4007   X. cxp 4767   -->wf 5368   X_cixp 6970

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-ixp 6971

This theorem is referenced by:  ixpexgg  6994