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Theorem xp1st 5792
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 4362 . 2  |-  ( A  e.  ( B  X.  C )  <->  E. b E. c ( A  = 
<. b ,  c >.  /\  ( b  e.  B  /\  c  e.  C
) ) )
2 vex 2560 . . . . . . 7  |-  b  e. 
_V
3 vex 2560 . . . . . . 7  |-  c  e. 
_V
42, 3op1std 5775 . . . . . 6  |-  ( A  =  <. b ,  c
>.  ->  ( 1st `  A
)  =  b )
54eleq1d 2106 . . . . 5  |-  ( A  =  <. b ,  c
>.  ->  ( ( 1st `  A )  e.  B  <->  b  e.  B ) )
65biimpar 281 . . . 4  |-  ( ( A  =  <. b ,  c >.  /\  b  e.  B )  ->  ( 1st `  A )  e.  B )
76adantrr 448 . . 3  |-  ( ( A  =  <. b ,  c >.  /\  (
b  e.  B  /\  c  e.  C )
)  ->  ( 1st `  A )  e.  B
)
87exlimivv 1776 . 2  |-  ( E. b E. c ( A  =  <. b ,  c >.  /\  (
b  e.  B  /\  c  e.  C )
)  ->  ( 1st `  A )  e.  B
)
91, 8sylbi 114 1  |-  ( A  e.  ( B  X.  C )  ->  ( 1st `  A )  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    = wceq 1243   E.wex 1381    e. wcel 1393   <.cop 3378    X. cxp 4343   ` cfv 4902   1stc1st 5765
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fv 4910  df-1st 5767
This theorem is referenced by:  dfplpq2  6452  dfmpq2  6453  enqbreq2  6455  enqdc1  6460  mulpipq2  6469  preqlu  6570  elnp1st2nd  6574  cauappcvgprlemladd  6756  elreal2  6907  cnref1o  8582  frecuzrdgrrn  9194
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