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Theorem xp2nd 6115
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.
StepHypRef Expression
1 elxp 4604 . 2  |-  ( A  e.  ( B  X.  C )  <->  E. b E. c ( A  = 
<. b ,  c >.  /\  ( b  e.  B  /\  c  e.  C
) ) )
2 vex 2715 . . . . . . 7  |-  b  e. 
_V
3 vex 2715 . . . . . . 7  |-  c  e. 
_V
42, 3op2ndd 6098 . . . . . 6  |-  ( A  =  <. b ,  c
>.  ->  ( 2nd `  A
)  =  c )
54eleq1d 2226 . . . . 5  |-  ( A  =  <. b ,  c
>.  ->  ( ( 2nd `  A )  e.  C  <->  c  e.  C ) )
65biimpar 295 . . . 4  |-  ( ( A  =  <. b ,  c >.  /\  c  e.  C )  ->  ( 2nd `  A )  e.  C )
76adantrl 470 . . 3  |-  ( ( A  =  <. b ,  c >.  /\  (
b  e.  B  /\  c  e.  C )
)  ->  ( 2nd `  A )  e.  C
)
87exlimivv 1876 . 2  |-  ( E. b E. c ( A  =  <. b ,  c >.  /\  (
b  e.  B  /\  c  e.  C )
)  ->  ( 2nd `  A )  e.  C
)
91, 8sylbi 120 1  |-  ( A  e.  ( B  X.  C )  ->  ( 2nd `  A )  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1335   E.wex 1472    e. wcel 2128   <.cop 3563    X. cxp 4585   ` cfv 5171   2ndc2nd 6088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170  ax-un 4394
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-br 3967  df-opab 4027  df-mpt 4028  df-id 4254  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-iota 5136  df-fun 5173  df-fv 5179  df-2nd 6090
This theorem is referenced by:  xpf1o  6790  xpmapenlem  6795  djuf1olem  6998  cc2lem  7187  dfplpq2  7275  dfmpq2  7276  enqbreq2  7278  enqdc1  7283  mulpipq2  7292  preqlu  7393  elnp1st2nd  7397  cauappcvgprlemladd  7579  elreal2  7751  cnref1o  9560  frecuzrdgrrn  10311  frec2uzrdg  10312  frecuzrdgrcl  10313  frecuzrdgtcl  10315  frecuzrdgsuc  10317  frecuzrdgrclt  10318  frecuzrdgg  10319  frecuzrdgdomlem  10320  frecuzrdgfunlem  10322  frecuzrdgsuctlem  10326  seq3val  10361  seqvalcd  10362  fisumcom2  11339  fprodcom2fi  11527  eucalgval  11935  eucalginv  11937  eucalglt  11938  eucalgcvga  11939  eucalg  11940  sqpweven  12054  2sqpwodd  12055  ctiunctlemudc  12208  tx1cn  12711  txdis  12719  txhmeo  12761  xmetxp  12949  xmetxpbl  12950  xmettxlem  12951  xmettx  12952
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