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Theorem xp1st 5918
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.
StepHypRef Expression
1 elxp 4445 . 2  |-  ( A  e.  ( B  X.  C )  <->  E. b E. c ( A  = 
<. b ,  c >.  /\  ( b  e.  B  /\  c  e.  C
) ) )
2 vex 2622 . . . . . . 7  |-  b  e. 
_V
3 vex 2622 . . . . . . 7  |-  c  e. 
_V
42, 3op1std 5901 . . . . . 6  |-  ( A  =  <. b ,  c
>.  ->  ( 1st `  A
)  =  b )
54eleq1d 2156 . . . . 5  |-  ( A  =  <. b ,  c
>.  ->  ( ( 1st `  A )  e.  B  <->  b  e.  B ) )
65biimpar 291 . . . 4  |-  ( ( A  =  <. b ,  c >.  /\  b  e.  B )  ->  ( 1st `  A )  e.  B )
76adantrr 463 . . 3  |-  ( ( A  =  <. b ,  c >.  /\  (
b  e.  B  /\  c  e.  C )
)  ->  ( 1st `  A )  e.  B
)
87exlimivv 1824 . 2  |-  ( E. b E. c ( A  =  <. b ,  c >.  /\  (
b  e.  B  /\  c  e.  C )
)  ->  ( 1st `  A )  e.  B
)
91, 8sylbi 119 1  |-  ( A  e.  ( B  X.  C )  ->  ( 1st `  A )  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289   E.wex 1426    e. wcel 1438   <.cop 3444    X. cxp 4426   ` cfv 5002   1stc1st 5891
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-iota 4967  df-fun 5004  df-fv 5010  df-1st 5893
This theorem is referenced by:  disjxp1  5983  xpf1o  6540  xpmapenlem  6545  djuf1olem  6724  djur  6736  eldju1st  6741  dfplpq2  6892  dfmpq2  6893  enqbreq2  6895  enqdc1  6900  mulpipq2  6909  preqlu  7010  elnp1st2nd  7014  cauappcvgprlemladd  7196  elreal2  7347  cnref1o  9102  frecuzrdgrrn  9780  frec2uzrdg  9781  frecuzrdgrcl  9782  frecuzrdgsuc  9786  frecuzrdgrclt  9787  frecuzrdgg  9788  frecuzrdgsuctlem  9795  iseqvalt  9838  seq3val  9839  fsum2dlemstep  10791  fisumcom2  10795  eucalgval  11129  eucalginv  11131  eucalglt  11132  eucialg  11134  sqpweven  11246  2sqpwodd  11247
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