Theorem xp1st 6374

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 4773	. 2 |- ( A e. ( B X. C ) <-> E. b E. c ( A = <. b , c >. /\ ( b e. B /\ c e. C ) ) )
2		vex 2818	. . . . . . 7 |- b e. _V
3		vex 2818	. . . . . . 7 |- c e. _V
4	2, 3	op1std 6357	. . . . . 6 |- ( A = <. b , c >. -> ( 1st ` A ) = b )
5	4	eleq1d 2303	. . . . 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 1948	. 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 1398   E.wex 1541   e. wcel 2205   <.cop 3698   X. cxp 4754   ` cfv 5359   1stc1st 6347

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-14 2208  ax-ext 2216  ax-sep 4234  ax-pow 4293  ax-pr 4328  ax-un 4560

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-br 4116  df-opab 4178  df-mpt 4179  df-id 4420  df-xp 4762  df-rel 4763  df-cnv 4764  df-co 4765  df-dm 4766  df-rn 4767  df-iota 5319  df-fun 5361  df-fv 5367  df-1st 6349

This theorem is referenced by:  disjxp1  6447  xpf1o  7112  xpmapenlem  7117  mapunen  7119  opabfi  7215  djuf1olem  7359  eldju1st  7377  exmidapne  7592  dfplpq2  7687  dfmpq2  7688  enqbreq2  7690  enqdc1  7695  mulpipq2  7704  preqlu  7805  elnp1st2nd  7809  cauappcvgprlemladd  7991  elreal2  8163  cnref1o  10006  frecuzrdgrrn  10799  frec2uzrdg  10800  frecuzrdgrcl  10801  frecuzrdgsuc  10805  frecuzrdgrclt  10806  frecuzrdgg  10807  frecuzrdgsuctlem  10814  seq3val  10851  seqvalcd  10852  fsum2dlemstep  12151  fisumcom2  12155  fprod2dlemstep  12339  fprodcom2fi  12343  eucalgval  12782  eucalginv  12784  eucalglt  12785  eucalg  12787  sqpweven  12903  2sqpwodd  12904  ctiunctlemudc  13278  xpsff1o  13619  tx2cn  15267  txdis  15274  txhmeo  15316  xmetxp  15504  xmetxpbl  15505  xmettxlem  15506  xmettx  15507  lgsquadlemofi  16081  lgsquadlem1  16082  lgsquadlem2  16083