Theorem unielxp 5820

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 5817	. 2 |- ( A e. ( B X. C ) <-> ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) )
2		elvvuni 4422	. . . 4 |- ( A e. ( _V X. _V ) -> U. A e. A )
3	2	adantr 270	. . 3 |- ( ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) -> U. A e. A )
4		simprl 497	. . . . . 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 2142	. . . . . . . 8 |- ( x = A -> ( U. A e. x <-> U. A e. A ) )
6		eleq1 2141	. . . . . . . . 9 |- ( x = A -> ( x e. ( _V X. _V ) <-> A e. ( _V X. _V ) ) )
7		fveq2 5198	. . . . . . . . . . 11 |- ( x = A -> ( 1st ` x ) = ( 1st ` A ) )
8	7	eleq1d 2147	. . . . . . . . . 10 |- ( x = A -> ( ( 1st ` x ) e. B <-> ( 1st ` A ) e. B ) )
9		fveq2 5198	. . . . . . . . . . 11 |- ( x = A -> ( 2nd ` x ) = ( 2nd ` A ) )
10	9	eleq1d 2147	. . . . . . . . . 10 |- ( x = A -> ( ( 2nd ` x ) e. C <-> ( 2nd ` A ) e. C ) )
11	8, 10	anbi12d 456	. . . . . . . . 9 |- ( x = A -> ( ( ( 1st ` x ) e. B /\ ( 2nd ` x ) e. C ) <-> ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) )
12	6, 11	anbi12d 456	. . . . . . . 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 456	. . . . . . 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 2686	. . . . . 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 3613	. . . . 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 132	. . . 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 5819	. . . . . 6 |- ( B X. C ) = { x e. ( _V X. _V ) | ( ( 1st ` x ) e. B /\ ( 2nd ` x ) e. C ) }
19		df-rab 2357	. . . . . 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 2101	. . . . 5 |- ( B X. C ) = { x | ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. B /\ ( 2nd ` x ) e. C ) ) }
21	20	unieqi 3611	. . . 4 |- U. ( B X. C ) = U. { x | ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. B /\ ( 2nd ` x ) e. C ) ) }
22	17, 21	syl6eleqr 2172	. . 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 413	. 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 119	1 |- ( A e. ( B X. C ) -> U. A e. U. ( B X. C ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   E.wex 1421   e. wcel 1433   {cab 2067   {crab 2352   _Vcvv 2601   U.cuni 3601   X. cxp 4361   ` cfv 4922   1stc1st 5785   2ndc2nd 5786

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fo 4928  df-fv 4930  df-1st 5787  df-2nd 5788

This theorem is referenced by: (None)