Theorem xp1st 6389

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 4786	. 2 |- ( A e. ( B X. C ) <-> E. b E. c ( A = <. b , c >. /\ ( b e. B /\ c e. C ) ) )
2		vex 2824	. . . . . . 7 |- b e. _V
3		vex 2824	. . . . . . 7 |- c e. _V
4	2, 3	op1std 6372	. . . . . 6 |- ( A = <. b , c >. -> ( 1st ` A ) = b )
5	4	eleq1d 2307	. . . . 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 483	. . 3 |- ( ( A = <. b , c >. /\ ( b e. B /\ c e. C ) ) -> ( 1st ` A ) e. B )
8	7	exlimivv 1952	. 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 1402   E.wex 1545   e. wcel 2209   <.cop 3708   X. cxp 4767   ` cfv 5372   1stc1st 6362

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fv 5380  df-1st 6364

This theorem is referenced by:  disjxp1  6462  xpf1o  7134  xpmapenlem  7139  mapunen  7141  opabfi  7237  djuf1olem  7383  eldju1st  7401  exmidapne  7616  dfplpq2  7711  dfmpq2  7712  enqbreq2  7714  enqdc1  7719  mulpipq2  7728  preqlu  7829  elnp1st2nd  7833  cauappcvgprlemladd  8015  elreal2  8187  cnref1o  10030  frecuzrdgrrn  10823  frec2uzrdg  10824  frecuzrdgrcl  10825  frecuzrdgsuc  10829  frecuzrdgrclt  10830  frecuzrdgg  10831  frecuzrdgsuctlem  10838  seq3val  10875  seqvalcd  10876  fsum2dlemstep  12179  fisumcom2  12183  fprod2dlemstep  12367  fprodcom2fi  12371  eucalgval  12810  eucalginv  12812  eucalglt  12813  eucalg  12815  sqpweven  12931  2sqpwodd  12932  ctiunctlemudc  13306  xpsff1o  13647  tx2cn  15294  txdis  15301  txhmeo  15343  xmetxp  15531  xmetxpbl  15532  xmettxlem  15533  xmettx  15534  lgsquadlemofi  16109  lgsquadlem1  16110  lgsquadlem2  16111