Theorem unielxp 6346

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 6342	. 2 |- ( A e. ( B X. C ) <-> ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) )
2		elvvuni 4796	. . . 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 531	. . . . . 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 2295	. . . . . . . 8 |- ( x = A -> ( U. A e. x <-> U. A e. A ) )
6		eleq1 2294	. . . . . . . . 9 |- ( x = A -> ( x e. ( _V X. _V ) <-> A e. ( _V X. _V ) ) )
7		fveq2 5648	. . . . . . . . . . 11 |- ( x = A -> ( 1st ` x ) = ( 1st ` A ) )
8	7	eleq1d 2300	. . . . . . . . . 10 |- ( x = A -> ( ( 1st ` x ) e. B <-> ( 1st ` A ) e. B ) )
9		fveq2 5648	. . . . . . . . . . 11 |- ( x = A -> ( 2nd ` x ) = ( 2nd ` A ) )
10	9	eleq1d 2300	. . . . . . . . . 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 2895	. . . . . 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 3910	. . . . 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 6345	. . . . . 6 |- ( B X. C ) = { x e. ( _V X. _V ) | ( ( 1st ` x ) e. B /\ ( 2nd ` x ) e. C ) }
19		df-rab 2520	. . . . . 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 2252	. . . . 5 |- ( B X. C ) = { x | ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. B /\ ( 2nd ` x ) e. C ) ) }
21	20	unieqi 3908	. . . 4 |- U. ( B X. C ) = U. { x | ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. B /\ ( 2nd ` x ) e. C ) ) }
22	17, 21	eleqtrrdi 2325	. . 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 1398   E.wex 1541   e. wcel 2202   {cab 2217   {crab 2515   _Vcvv 2803   U.cuni 3898   X. cxp 4729   ` cfv 5333   1stc1st 6310   2ndc2nd 6311

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fo 5339  df-fv 5341  df-1st 6312  df-2nd 6313

This theorem is referenced by: (None)