Theorem xp1st 5974

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 4484	. 2 |- ( A e. ( B X. C ) <-> E. b E. c ( A = <. b , c >. /\ ( b e. B /\ c e. C ) ) )
2		vex 2636	. . . . . . 7 |- b e. _V
3		vex 2636	. . . . . . 7 |- c e. _V
4	2, 3	op1std 5957	. . . . . 6 |- ( A = <. b , c >. -> ( 1st ` A ) = b )
5	4	eleq1d 2163	. . . . 5 |- ( A = <. b , c >. -> ( ( 1st ` A ) e. B <-> b e. B ) )
6	5	biimpar 292	. . . 4 |- ( ( A = <. b , c >. /\ b e. B ) -> ( 1st ` A ) e. B )
7	6	adantrr 464	. . 3 |- ( ( A = <. b , c >. /\ ( b e. B /\ c e. C ) ) -> ( 1st ` A ) e. B )
8	7	exlimivv 1831	. 2 |- ( E. b E. c ( A = <. b , c >. /\ ( b e. B /\ c e. C ) ) -> ( 1st ` A ) e. B )
9	1, 8	sylbi 120	1 |- ( A e. ( B X. C ) -> ( 1st ` A ) e. B )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1296   E.wex 1433   e. wcel 1445   <.cop 3469   X. cxp 4465   ` cfv 5049   1stc1st 5947

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-iota 5014  df-fun 5051  df-fv 5057  df-1st 5949

This theorem is referenced by:  disjxp1  6039  xpf1o  6640  xpmapenlem  6645  djuf1olem  6825  djur  6837  eldju1st  6842  dfplpq2  7010  dfmpq2  7011  enqbreq2  7013  enqdc1  7018  mulpipq2  7027  preqlu  7128  elnp1st2nd  7132  cauappcvgprlemladd  7314  elreal2  7465  cnref1o  9232  frecuzrdgrrn  9964  frec2uzrdg  9965  frecuzrdgrcl  9966  frecuzrdgsuc  9970  frecuzrdgrclt  9971  frecuzrdgg  9972  frecuzrdgsuctlem  9979  iseqvalt  10019  seq3val  10020  fsum2dlemstep  10992  fisumcom2  10996  eucalgval  11478  eucalginv  11480  eucalglt  11481  eucalg  11483  sqpweven  11595  2sqpwodd  11596