Theorem unielxp 6153

Description: The membership relation for a cross product is inherited by union. (Contributed by NM, 16-Sep-2006.)

Assertion

Ref	Expression
unielxp	|- ( A e. ( B X. C ) -> U. A e. U. ( B X. C ) )

Proof of Theorem unielxp

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elxp7 6149	. 2 |- ( A e. ( B X. C ) <-> ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) )
2		elvvuni 4675	. . . 4 |- ( A e. ( _V X. _V ) -> U. A e. A )
3	2	adantr 274	. . 3 |- ( ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) -> U. A e. A )
4		simprl 526	. . . . . 6 |- ( ( U. A e. A /\ ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) ) -> A e. ( _V X. _V ) )
5		eleq2 2234	. . . . . . . 8 |- ( x = A -> ( U. A e. x <-> U. A e. A ) )
6		eleq1 2233	. . . . . . . . 9 |- ( x = A -> ( x e. ( _V X. _V ) <-> A e. ( _V X. _V ) ) )
7		fveq2 5496	. . . . . . . . . . 11 |- ( x = A -> ( 1st ` x ) = ( 1st ` A ) )
8	7	eleq1d 2239	. . . . . . . . . 10 |- ( x = A -> ( ( 1st ` x ) e. B <-> ( 1st ` A ) e. B ) )
9		fveq2 5496	. . . . . . . . . . 11 |- ( x = A -> ( 2nd ` x ) = ( 2nd ` A ) )
10	9	eleq1d 2239	. . . . . . . . . 10 |- ( x = A -> ( ( 2nd ` x ) e. C <-> ( 2nd ` A ) e. C ) )
11	8, 10	anbi12d 470	. . . . . . . . 9 |- ( x = A -> ( ( ( 1st ` x ) e. B /\ ( 2nd ` x ) e. C ) <-> ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) )
12	6, 11	anbi12d 470	. . . . . . . 8 |- ( x = A -> ( ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. B /\ ( 2nd ` x ) e. C ) ) <-> ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) ) )
13	5, 12	anbi12d 470	. . . . . . 7 |- ( x = A -> ( ( U. A e. x /\ ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. B /\ ( 2nd ` x ) e. C ) ) ) <-> ( U. A e. A /\ ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) ) ) )
14	13	spcegv 2818	. . . . . 6 |- ( A e. ( _V X. _V ) -> ( ( U. A e. A /\ ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) ) -> E. x ( U. A e. x /\ ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. B /\ ( 2nd ` x ) e. C ) ) ) ) )
15	4, 14	mpcom 36	. . . . 5 |- ( ( U. A e. A /\ ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) ) -> E. x ( U. A e. x /\ ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. B /\ ( 2nd ` x ) e. C ) ) ) )
16		eluniab 3808	. . . . 5 |- ( U. A e. U. { x | ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. B /\ ( 2nd ` x ) e. C ) ) } <-> E. x ( U. A e. x /\ ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. B /\ ( 2nd ` x ) e. C ) ) ) )
17	15, 16	sylibr 133	. . . 4 |- ( ( U. A e. A /\ ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) ) -> U. A e. U. { x | ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. B /\ ( 2nd ` x ) e. C ) ) } )
18		xp2 6152	. . . . . 6 |- ( B X. C ) = { x e. ( _V X. _V ) | ( ( 1st ` x ) e. B /\ ( 2nd ` x ) e. C ) }
19		df-rab 2457	. . . . . 6 |- { x e. ( _V X. _V ) | ( ( 1st ` x ) e. B /\ ( 2nd ` x ) e. C ) } = { x | ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. B /\ ( 2nd ` x ) e. C ) ) }
20	18, 19	eqtri 2191	. . . . 5 |- ( B X. C ) = { x | ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. B /\ ( 2nd ` x ) e. C ) ) }
21	20	unieqi 3806	. . . 4 |- U. ( B X. C ) = U. { x | ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. B /\ ( 2nd ` x ) e. C ) ) }
22	17, 21	eleqtrrdi 2264	. . 3 |- ( ( U. A e. A /\ ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) ) -> U. A e. U. ( B X. C ) )
23	3, 22	mpancom 420	. 2 |- ( ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) -> U. A e. U. ( B X. C ) )
24	1, 23	sylbi 120	1 |- ( A e. ( B X. C ) -> U. A e. U. ( B X. C ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   E.wex 1485   e. wcel 2141   {cab 2156   {crab 2452   _Vcvv 2730   U.cuni 3796   X. cxp 4609   ` cfv 5198   1stc1st 6117   2ndc2nd 6118

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-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

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-rab 2457  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-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-fo 5204  df-fv 5206  df-1st 6119  df-2nd 6120

This theorem is referenced by: (None)