Theorem xp1st 6359

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 4766	. 2 |- ( A e. ( B X. C ) <-> E. b E. c ( A = <. b , c >. /\ ( b e. B /\ c e. C ) ) )
2		vex 2816	. . . . . . 7 |- b e. _V
3		vex 2816	. . . . . . 7 |- c e. _V
4	2, 3	op1std 6342	. . . . . 6 |- ( A = <. b , c >. -> ( 1st ` A ) = b )
5	4	eleq1d 2301	. . . . 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 1946	. 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 2203   <.cop 3692   X. cxp 4747   ` cfv 5352   1stc1st 6332

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-iota 5312  df-fun 5354  df-fv 5360  df-1st 6334

This theorem is referenced by:  disjxp1  6432  xpf1o  7097  xpmapenlem  7102  mapunen  7104  opabfi  7200  djuf1olem  7344  eldju1st  7362  exmidapne  7574  dfplpq2  7669  dfmpq2  7670  enqbreq2  7672  enqdc1  7677  mulpipq2  7686  preqlu  7787  elnp1st2nd  7791  cauappcvgprlemladd  7973  elreal2  8145  cnref1o  9983  frecuzrdgrrn  10770  frec2uzrdg  10771  frecuzrdgrcl  10772  frecuzrdgsuc  10776  frecuzrdgrclt  10777  frecuzrdgg  10778  frecuzrdgsuctlem  10785  seq3val  10822  seqvalcd  10823  fsum2dlemstep  12120  fisumcom2  12124  fprod2dlemstep  12308  fprodcom2fi  12312  eucalgval  12751  eucalginv  12753  eucalglt  12754  eucalg  12756  sqpweven  12872  2sqpwodd  12873  ctiunctlemudc  13188  xpsff1o  13562  tx2cn  15135  txdis  15142  txhmeo  15184  xmetxp  15372  xmetxpbl  15373  xmettxlem  15374  xmettx  15375  lgsquadlemofi  15949  lgsquadlem1  15950  lgsquadlem2  15951