Theorem xp2nd 6373

Description: Location of the second element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
xp2nd	|- ( A e. ( B X. C ) -> ( 2nd ` A ) e. C )

Proof of Theorem xp2nd

Dummy variables b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elxp 4771	. 2 |- ( A e. ( B X. C ) <-> E. b E. c ( A = <. b , c >. /\ ( b e. B /\ c e. C ) ) )
2		vex 2818	. . . . . . 7 |- b e. _V
3		vex 2818	. . . . . . 7 |- c e. _V
4	2, 3	op2ndd 6356	. . . . . 6 |- ( A = <. b , c >. -> ( 2nd ` A ) = c )
5	4	eleq1d 2303	. . . . 5 |- ( A = <. b , c >. -> ( ( 2nd ` A ) e. C <-> c e. C ) )
6	5	biimpar 297	. . . 4 |- ( ( A = <. b , c >. /\ c e. C ) -> ( 2nd ` A ) e. C )
7	6	adantrl 478	. . 3 |- ( ( A = <. b , c >. /\ ( b e. B /\ c e. C ) ) -> ( 2nd ` A ) e. C )
8	7	exlimivv 1948	. 2 |- ( E. b E. c ( A = <. b , c >. /\ ( b e. B /\ c e. C ) ) -> ( 2nd ` A ) e. C )
9	1, 8	sylbi 121	1 |- ( A e. ( B X. C ) -> ( 2nd ` A ) e. C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   E.wex 1541   e. wcel 2205   <.cop 3697   X. cxp 4752   ` cfv 5357   2ndc2nd 6346

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-iota 5317  df-fun 5359  df-fv 5365  df-2nd 6348

This theorem is referenced by:  xpf1o  7110  xpmapenlem  7115  mapunen  7117  opabfi  7213  djuf1olem  7357  exmidapne  7590  cc2lem  7596  dfplpq2  7685  dfmpq2  7686  enqbreq2  7688  enqdc1  7693  mulpipq2  7702  preqlu  7803  elnp1st2nd  7807  cauappcvgprlemladd  7989  elreal2  8161  cnref1o  10001  frecuzrdgrrn  10794  frec2uzrdg  10795  frecuzrdgrcl  10796  frecuzrdgtcl  10798  frecuzrdgsuc  10800  frecuzrdgrclt  10801  frecuzrdgg  10802  frecuzrdgdomlem  10803  frecuzrdgfunlem  10805  frecuzrdgsuctlem  10809  seq3val  10846  seqvalcd  10847  fisumcom2  12149  fprodcom2fi  12337  eucalgval  12776  eucalginv  12778  eucalglt  12779  eucalgcvga  12780  eucalg  12781  sqpweven  12897  2sqpwodd  12898  ctiunctlemudc  13272  xpsff1o  13613  tx1cn  15260  txdis  15268  txhmeo  15310  xmetxp  15498  xmetxpbl  15499  xmettxlem  15500  xmettx  15501