Theorem uniixp 6678

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 6677	. . . . 5 |- ( f e. X_ x e. A B -> f : A --> U_ x e. A B )
2		fssxp 5349	. . . . 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 3560	. . . 4 |- ( f e. ~P ( A X. U_ x e. A B ) <-> f C_ ( A X. U_ x e. A B ) )
5	3, 4	sylibr 133	. . 3 |- ( f e. X_ x e. A B -> f e. ~P ( A X. U_ x e. A B ) )
6	5	ssriv 3141	. 2 |- X_ x e. A B C_ ~P ( A X. U_ x e. A B )
7		sspwuni 3944	. 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 144	1 |- U. X_ x e. A B C_ ( A X. U_ x e. A B )

Syntax hints:   e. wcel 2135   C_ wss 3111   ~Pcpw 3553   U.cuni 3783   U_ciun 3860   X. cxp 4596   -->wf 5178   X_cixp 6655

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2723  df-sbc 2947  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-id 4265  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-fv 5190  df-ixp 6656

This theorem is referenced by:  ixpexgg  6679