Theorem uniixp 6699

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 6698	. . . . 5 |- ( f e. X_ x e. A B -> f : A --> U_ x e. A B )
2		fssxp 5365	. . . . 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 3573	. . . 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 3151	. 2 |- X_ x e. A B C_ ~P ( A X. U_ x e. A B )
7		sspwuni 3957	. 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 2141   C_ wss 3121   ~Pcpw 3566   U.cuni 3796   U_ciun 3873   X. cxp 4609   -->wf 5194   X_cixp 6676

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-ixp 6677

This theorem is referenced by:  ixpexgg  6700