Theorem xp1st 6337

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 4748	. 2 |- ( A e. ( B X. C ) <-> E. b E. c ( A = <. b , c >. /\ ( b e. B /\ c e. C ) ) )
2		vex 2806	. . . . . . 7 |- b e. _V
3		vex 2806	. . . . . . 7 |- c e. _V
4	2, 3	op1std 6320	. . . . . 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 1398   E.wex 1541   e. wcel 2202   <.cop 3676   X. cxp 4729   ` cfv 5333   1stc1st 6310

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-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fv 5341  df-1st 6312

This theorem is referenced by:  disjxp1  6410  xpf1o  7073  xpmapenlem  7078  opabfi  7175  djuf1olem  7295  eldju1st  7313  exmidapne  7522  dfplpq2  7617  dfmpq2  7618  enqbreq2  7620  enqdc1  7625  mulpipq2  7634  preqlu  7735  elnp1st2nd  7739  cauappcvgprlemladd  7921  elreal2  8093  cnref1o  9929  frecuzrdgrrn  10716  frec2uzrdg  10717  frecuzrdgrcl  10718  frecuzrdgsuc  10722  frecuzrdgrclt  10723  frecuzrdgg  10724  frecuzrdgsuctlem  10731  seq3val  10768  seqvalcd  10769  fsum2dlemstep  12058  fisumcom2  12062  fprod2dlemstep  12246  fprodcom2fi  12250  eucalgval  12689  eucalginv  12691  eucalglt  12692  eucalg  12694  sqpweven  12810  2sqpwodd  12811  ctiunctlemudc  13121  xpsff1o  13495  tx2cn  15064  txdis  15071  txhmeo  15113  xmetxp  15301  xmetxpbl  15302  xmettxlem  15303  xmettx  15304  lgsquadlemofi  15878  lgsquadlem1  15879  lgsquadlem2  15880