Theorem unielxp 6332

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 6328	. 2 |- ( A e. ( B X. C ) <-> ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) )
2		elvvuni 4788	. . . 4 |- ( A e. ( _V X. _V ) -> U. A e. A )
3	2	adantr 276	. . 3 |- ( ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) -> U. A e. A )
4		simprl 529	. . . . . 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 2293	. . . . . . . 8 |- ( x = A -> ( U. A e. x <-> U. A e. A ) )
6		eleq1 2292	. . . . . . . . 9 |- ( x = A -> ( x e. ( _V X. _V ) <-> A e. ( _V X. _V ) ) )
7		fveq2 5635	. . . . . . . . . . 11 |- ( x = A -> ( 1st ` x ) = ( 1st ` A ) )
8	7	eleq1d 2298	. . . . . . . . . 10 |- ( x = A -> ( ( 1st ` x ) e. B <-> ( 1st ` A ) e. B ) )
9		fveq2 5635	. . . . . . . . . . 11 |- ( x = A -> ( 2nd ` x ) = ( 2nd ` A ) )
10	9	eleq1d 2298	. . . . . . . . . 10 |- ( x = A -> ( ( 2nd ` x ) e. C <-> ( 2nd ` A ) e. C ) )
11	8, 10	anbi12d 473	. . . . . . . . 9 |- ( x = A -> ( ( ( 1st ` x ) e. B /\ ( 2nd ` x ) e. C ) <-> ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) )
12	6, 11	anbi12d 473	. . . . . . . 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 473	. . . . . . 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 2892	. . . . . 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 3903	. . . . 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 134	. . . 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 6331	. . . . . 6 |- ( B X. C ) = { x e. ( _V X. _V ) | ( ( 1st ` x ) e. B /\ ( 2nd ` x ) e. C ) }
19		df-rab 2517	. . . . . 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 2250	. . . . 5 |- ( B X. C ) = { x | ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. B /\ ( 2nd ` x ) e. C ) ) }
21	20	unieqi 3901	. . . 4 |- U. ( B X. C ) = U. { x | ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. B /\ ( 2nd ` x ) e. C ) ) }
22	17, 21	eleqtrrdi 2323	. . 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 422	. 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 121	1 |- ( A e. ( B X. C ) -> U. A e. U. ( B X. C ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   E.wex 1538   e. wcel 2200   {cab 2215   {crab 2512   _Vcvv 2800   U.cuni 3891   X. cxp 4721   ` cfv 5324   1stc1st 6296   2ndc2nd 6297

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fo 5330  df-fv 5332  df-1st 6298  df-2nd 6299

This theorem is referenced by: (None)