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Theorem op1stg 6691
Description: Extract the first member of an ordered pair. (Contributed by NM, 19-Jul-2005.)
Assertion
Ref Expression
op1stg  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 1st `  <. A ,  B >. )  =  A )

Proof of Theorem op1stg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opeq1 4159 . . . 4  |-  ( x  =  A  ->  <. x ,  y >.  =  <. A ,  y >. )
21fveq2d 5795 . . 3  |-  ( x  =  A  ->  ( 1st `  <. x ,  y
>. )  =  ( 1st `  <. A ,  y
>. ) )
3 id 22 . . 3  |-  ( x  =  A  ->  x  =  A )
42, 3eqeq12d 2473 . 2  |-  ( x  =  A  ->  (
( 1st `  <. x ,  y >. )  =  x  <->  ( 1st `  <. A ,  y >. )  =  A ) )
5 opeq2 4160 . . . 4  |-  ( y  =  B  ->  <. A , 
y >.  =  <. A ,  B >. )
65fveq2d 5795 . . 3  |-  ( y  =  B  ->  ( 1st `  <. A ,  y
>. )  =  ( 1st `  <. A ,  B >. ) )
76eqeq1d 2453 . 2  |-  ( y  =  B  ->  (
( 1st `  <. A ,  y >. )  =  A  <->  ( 1st `  <. A ,  B >. )  =  A ) )
8 vex 3073 . . 3  |-  x  e. 
_V
9 vex 3073 . . 3  |-  y  e. 
_V
108, 9op1st 6687 . 2  |-  ( 1st `  <. x ,  y
>. )  =  x
114, 7, 10vtocl2g 3132 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 1st `  <. A ,  B >. )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   <.cop 3983   ` cfv 5518   1stc1st 6677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-iota 5481  df-fun 5520  df-fv 5526  df-1st 6679
This theorem is referenced by:  ot1stg  6693  ot2ndg  6694  1stconst  6763  curry2  6769  mpt2xopn0yelv  6832  mpt2xopoveq  6838  xpmapenlem  7580  fpwwe  8916  addpipq  9209  mulpipq  9212  ordpipq  9214  swrdval  12417  ruclem1  13617  qnumdenbi  13926  oppccofval  14759  funcf2  14882  cofuval2  14901  resfval2  14907  resf1st  14908  isnat  14961  fucco  14976  homadm  15012  setcco  15055  xpcco  15097  xpchom2  15100  xpcco2  15101  evlf2  15132  curfval  15137  curf1cl  15142  uncf1  15150  uncf2  15151  diag11  15157  diag12  15158  diag2  15159  hof2fval  15169  yonedalem21  15187  yonedalem22  15192  mvmulfval  18466  imasdsf1olem  20066  ovolicc1  21117  ioombl1lem3  21159  ioombl1lem4  21160  brcgr  23283  nbgraop  23472  rngoablo2  24046  vcoprne  24094  fgreu  26126  fvtransport  28199  mpt2sn  30863  bj-inftyexpiinv  32839  bj-finsumval0  32891  dvhopvadd  35046  dvhopvsca  35055  dvhopaddN  35067  dvhopspN  35068
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