Theorem xp1st 6309

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 4735	. 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 6292	. . . . . 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 4716   ` cfv 5317   1stc1st 6282

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 4201  ax-pow 4257  ax-pr 4292  ax-un 4523

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 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-iota 5277  df-fun 5319  df-fv 5325  df-1st 6284

This theorem is referenced by:  disjxp1  6380  xpf1o  7001  xpmapenlem  7006  opabfi  7096  djuf1olem  7216  eldju1st  7234  exmidapne  7442  dfplpq2  7537  dfmpq2  7538  enqbreq2  7540  enqdc1  7545  mulpipq2  7554  preqlu  7655  elnp1st2nd  7659  cauappcvgprlemladd  7841  elreal2  8013  cnref1o  9842  frecuzrdgrrn  10625  frec2uzrdg  10626  frecuzrdgrcl  10627  frecuzrdgsuc  10631  frecuzrdgrclt  10632  frecuzrdgg  10633  frecuzrdgsuctlem  10640  seq3val  10677  seqvalcd  10678  fsum2dlemstep  11940  fisumcom2  11944  fprod2dlemstep  12128  fprodcom2fi  12132  eucalgval  12571  eucalginv  12573  eucalglt  12574  eucalg  12576  sqpweven  12692  2sqpwodd  12693  ctiunctlemudc  13003  xpsff1o  13377  tx2cn  14938  txdis  14945  txhmeo  14987  xmetxp  15175  xmetxpbl  15176  xmettxlem  15177  xmettx  15178  lgsquadlemofi  15749  lgsquadlem1  15750  lgsquadlem2  15751