Theorem xp1st 6317

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 4736	. 2 |- ( A e. ( B X. C ) <-> E. b E. c ( A = <. b , c >. /\ ( b e. B /\ c e. C ) ) )
2		vex 2802	. . . . . . 7 |- b e. _V
3		vex 2802	. . . . . . 7 |- c e. _V
4	2, 3	op1std 6300	. . . . . 6 |- ( A = <. b , c >. -> ( 1st ` A ) = b )
5	4	eleq1d 2298	. . . . 5 |- ( A = <. b , c >. -> ( ( 1st ` A ) e. B <-> b e. B ) )
6	5	biimpar 297	. . . 4 |- ( ( A = <. b , c >. /\ b e. B ) -> ( 1st ` A ) e. B )
7	6	adantrr 479	. . 3 |- ( ( A = <. b , c >. /\ ( b e. B /\ c e. C ) ) -> ( 1st ` A ) e. B )
8	7	exlimivv 1943	. 2 |- ( E. b E. c ( A = <. b , c >. /\ ( b e. B /\ c e. C ) ) -> ( 1st ` A ) e. B )
9	1, 8	sylbi 121	1 |- ( A e. ( B X. C ) -> ( 1st ` A ) e. B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   E.wex 1538   e. wcel 2200   <.cop 3669   X. cxp 4717   ` cfv 5318   1stc1st 6290

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 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

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-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fv 5326  df-1st 6292

This theorem is referenced by:  disjxp1  6388  xpf1o  7013  xpmapenlem  7018  opabfi  7111  djuf1olem  7231  eldju1st  7249  exmidapne  7457  dfplpq2  7552  dfmpq2  7553  enqbreq2  7555  enqdc1  7560  mulpipq2  7569  preqlu  7670  elnp1st2nd  7674  cauappcvgprlemladd  7856  elreal2  8028  cnref1o  9858  frecuzrdgrrn  10642  frec2uzrdg  10643  frecuzrdgrcl  10644  frecuzrdgsuc  10648  frecuzrdgrclt  10649  frecuzrdgg  10650  frecuzrdgsuctlem  10657  seq3val  10694  seqvalcd  10695  fsum2dlemstep  11960  fisumcom2  11964  fprod2dlemstep  12148  fprodcom2fi  12152  eucalgval  12591  eucalginv  12593  eucalglt  12594  eucalg  12596  sqpweven  12712  2sqpwodd  12713  ctiunctlemudc  13023  xpsff1o  13397  tx2cn  14959  txdis  14966  txhmeo  15008  xmetxp  15196  xmetxpbl  15197  xmettxlem  15198  xmettx  15199  lgsquadlemofi  15770  lgsquadlem1  15771  lgsquadlem2  15772