Theorem xp1st 6323

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 4740	. 2 |- ( A e. ( B X. C ) <-> E. b E. c ( A = <. b , c >. /\ ( b e. B /\ c e. C ) ) )
2		vex 2803	. . . . . . 7 |- b e. _V
3		vex 2803	. . . . . . 7 |- c e. _V
4	2, 3	op1std 6306	. . . . . 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 3670   X. cxp 4721   ` cfv 5324   1stc1st 6296

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 4205  ax-pow 4262  ax-pr 4297  ax-un 4528

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 2802  df-sbc 3030  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-iota 5284  df-fun 5326  df-fv 5332  df-1st 6298

This theorem is referenced by:  disjxp1  6396  xpf1o  7025  xpmapenlem  7030  opabfi  7123  djuf1olem  7243  eldju1st  7261  exmidapne  7469  dfplpq2  7564  dfmpq2  7565  enqbreq2  7567  enqdc1  7572  mulpipq2  7581  preqlu  7682  elnp1st2nd  7686  cauappcvgprlemladd  7868  elreal2  8040  cnref1o  9875  frecuzrdgrrn  10660  frec2uzrdg  10661  frecuzrdgrcl  10662  frecuzrdgsuc  10666  frecuzrdgrclt  10667  frecuzrdgg  10668  frecuzrdgsuctlem  10675  seq3val  10712  seqvalcd  10713  fsum2dlemstep  11985  fisumcom2  11989  fprod2dlemstep  12173  fprodcom2fi  12177  eucalgval  12616  eucalginv  12618  eucalglt  12619  eucalg  12621  sqpweven  12737  2sqpwodd  12738  ctiunctlemudc  13048  xpsff1o  13422  tx2cn  14984  txdis  14991  txhmeo  15033  xmetxp  15221  xmetxpbl  15222  xmettxlem  15223  xmettx  15224  lgsquadlemofi  15795  lgsquadlem1  15796  lgsquadlem2  15797