Theorem xp2nd 6329

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 4742	. 2 |- ( A e. ( B X. C ) <-> E. b E. c ( A = <. b , c >. /\ ( b e. B /\ c e. C ) ) )
2		vex 2805	. . . . . . 7 |- b e. _V
3		vex 2805	. . . . . . 7 |- c e. _V
4	2, 3	op2ndd 6312	. . . . . 6 |- ( A = <. b , c >. -> ( 2nd ` A ) = c )
5	4	eleq1d 2300	. . . . 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 1945	. 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 1397   E.wex 1540   e. wcel 2202   <.cop 3672   X. cxp 4723   ` cfv 5326   2ndc2nd 6302

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fv 5334  df-2nd 6304

This theorem is referenced by:  xpf1o  7030  xpmapenlem  7035  opabfi  7132  djuf1olem  7252  exmidapne  7479  cc2lem  7485  dfplpq2  7574  dfmpq2  7575  enqbreq2  7577  enqdc1  7582  mulpipq2  7591  preqlu  7692  elnp1st2nd  7696  cauappcvgprlemladd  7878  elreal2  8050  cnref1o  9885  frecuzrdgrrn  10670  frec2uzrdg  10671  frecuzrdgrcl  10672  frecuzrdgtcl  10674  frecuzrdgsuc  10676  frecuzrdgrclt  10677  frecuzrdgg  10678  frecuzrdgdomlem  10679  frecuzrdgfunlem  10681  frecuzrdgsuctlem  10685  seq3val  10722  seqvalcd  10723  fisumcom2  12000  fprodcom2fi  12188  eucalgval  12627  eucalginv  12629  eucalglt  12630  eucalgcvga  12631  eucalg  12632  sqpweven  12748  2sqpwodd  12749  ctiunctlemudc  13059  xpsff1o  13433  tx1cn  14995  txdis  15003  txhmeo  15045  xmetxp  15233  xmetxpbl  15234  xmettxlem  15235  xmettx  15236