Theorem unielxp 6408

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 6404	. 2 |- ( A e. ( B X. C ) <-> ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) )
2		elvvuni 4839	. . . 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 535	. . . . . 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 2302	. . . . . . . 8 |- ( x = A -> ( U. A e. x <-> U. A e. A ) )
6		eleq1 2301	. . . . . . . . 9 |- ( x = A -> ( x e. ( _V X. _V ) <-> A e. ( _V X. _V ) ) )
7		fveq2 5695	. . . . . . . . . . 11 |- ( x = A -> ( 1st ` x ) = ( 1st ` A ) )
8	7	eleq1d 2307	. . . . . . . . . 10 |- ( x = A -> ( ( 1st ` x ) e. B <-> ( 1st ` A ) e. B ) )
9		fveq2 5695	. . . . . . . . . . 11 |- ( x = A -> ( 2nd ` x ) = ( 2nd ` A ) )
10	9	eleq1d 2307	. . . . . . . . . 10 |- ( x = A -> ( ( 2nd ` x ) e. C <-> ( 2nd ` A ) e. C ) )
11	8, 10	anbi12d 477	. . . . . . . . 9 |- ( x = A -> ( ( ( 1st ` x ) e. B /\ ( 2nd ` x ) e. C ) <-> ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) )
12	6, 11	anbi12d 477	. . . . . . . 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 477	. . . . . . 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 2913	. . . . . 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 3947	. . . . 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 6407	. . . . . 6 |- ( B X. C ) = { x e. ( _V X. _V ) | ( ( 1st ` x ) e. B /\ ( 2nd ` x ) e. C ) }
19		df-rab 2537	. . . . . 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 2259	. . . . 5 |- ( B X. C ) = { x | ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. B /\ ( 2nd ` x ) e. C ) ) }
21	20	unieqi 3945	. . . 4 |- U. ( B X. C ) = U. { x | ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. B /\ ( 2nd ` x ) e. C ) ) }
22	17, 21	eleqtrrdi 2332	. . 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 426	. 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   E.wex 1545   e. wcel 2209   {cab 2224   {crab 2532   _Vcvv 2821   U.cuni 3935   X. cxp 4772   ` cfv 5377   1stc1st 6372   2ndc2nd 6373

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fo 5383  df-fv 5385  df-1st 6374  df-2nd 6375

This theorem is used by: (None)