Theorem xp1st 6056

Description: Location of the first element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
xp1st	|- ( A e. ( B X. C ) -> ( 1st ` A ) e. B )

Proof of Theorem xp1st

Dummy variables b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elxp 4551	. 2 |- ( A e. ( B X. C ) <-> E. b E. c ( A = <. b , c >. /\ ( b e. B /\ c e. C ) ) )
2		vex 2684	. . . . . . 7 |- b e. _V
3		vex 2684	. . . . . . 7 |- c e. _V
4	2, 3	op1std 6039	. . . . . 6 |- ( A = <. b , c >. -> ( 1st ` A ) = b )
5	4	eleq1d 2206	. . . . 5 |- ( A = <. b , c >. -> ( ( 1st ` A ) e. B <-> b e. B ) )
6	5	biimpar 295	. . . 4 |- ( ( A = <. b , c >. /\ b e. B ) -> ( 1st ` A ) e. B )
7	6	adantrr 470	. . 3 |- ( ( A = <. b , c >. /\ ( b e. B /\ c e. C ) ) -> ( 1st ` A ) e. B )
8	7	exlimivv 1868	. 2 |- ( E. b E. c ( A = <. b , c >. /\ ( b e. B /\ c e. C ) ) -> ( 1st ` A ) e. B )
9	1, 8	sylbi 120	1 |- ( A e. ( B X. C ) -> ( 1st ` A ) e. B )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   E.wex 1468   e. wcel 1480   <.cop 3525   X. cxp 4532   ` cfv 5118   1stc1st 6029

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fv 5126  df-1st 6031

This theorem is referenced by:  disjxp1  6126  xpf1o  6731  xpmapenlem  6736  djuf1olem  6931  eldju1st  6949  dfplpq2  7155  dfmpq2  7156  enqbreq2  7158  enqdc1  7163  mulpipq2  7172  preqlu  7273  elnp1st2nd  7277  cauappcvgprlemladd  7459  elreal2  7631  cnref1o  9433  frecuzrdgrrn  10174  frec2uzrdg  10175  frecuzrdgrcl  10176  frecuzrdgsuc  10180  frecuzrdgrclt  10181  frecuzrdgg  10182  frecuzrdgsuctlem  10189  seq3val  10224  seqvalcd  10225  fsum2dlemstep  11196  fisumcom2  11200  eucalgval  11724  eucalginv  11726  eucalglt  11727  eucalg  11729  sqpweven  11842  2sqpwodd  11843  ctiunctlemudc  11939  tx2cn  12428  txdis  12435  txhmeo  12477  xmetxp  12665  xmetxpbl  12666  xmettxlem  12667  xmettx  12668