Theorem xp1st 6327

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 4742	. 2 |- ( A e. ( B X. C ) <-> E. b E. c ( A = <. b , c >. /\ ( b e. B /\ c e. C ) ) )
2		vex 2805	. . . . . . 7 |- b e. _V
3		vex 2805	. . . . . . 7 |- c e. _V
4	2, 3	op1std 6310	. . . . . 6 |- ( A = <. b , c >. -> ( 1st ` A ) = b )
5	4	eleq1d 2300	. . . . 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 1945	. 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 1397   E.wex 1540   e. wcel 2202   <.cop 3672   X. cxp 4723   ` cfv 5326   1stc1st 6300

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fv 5334  df-1st 6302

This theorem is referenced by:  disjxp1  6400  xpf1o  7029  xpmapenlem  7034  opabfi  7131  djuf1olem  7251  eldju1st  7269  exmidapne  7478  dfplpq2  7573  dfmpq2  7574  enqbreq2  7576  enqdc1  7581  mulpipq2  7590  preqlu  7691  elnp1st2nd  7695  cauappcvgprlemladd  7877  elreal2  8049  cnref1o  9884  frecuzrdgrrn  10669  frec2uzrdg  10670  frecuzrdgrcl  10671  frecuzrdgsuc  10675  frecuzrdgrclt  10676  frecuzrdgg  10677  frecuzrdgsuctlem  10684  seq3val  10721  seqvalcd  10722  fsum2dlemstep  11994  fisumcom2  11998  fprod2dlemstep  12182  fprodcom2fi  12186  eucalgval  12625  eucalginv  12627  eucalglt  12628  eucalg  12630  sqpweven  12746  2sqpwodd  12747  ctiunctlemudc  13057  xpsff1o  13431  tx2cn  14993  txdis  15000  txhmeo  15042  xmetxp  15230  xmetxpbl  15231  xmettxlem  15232  xmettx  15233  lgsquadlemofi  15804  lgsquadlem1  15805  lgsquadlem2  15806