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Theorem op1stg 6148
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 3812 . . . 4  |-  ( x  =  A  ->  <. x ,  y >.  =  <. A ,  y >. )
21fveq2d 5545 . . 3  |-  ( x  =  A  ->  ( 1st `  <. x ,  y
>. )  =  ( 1st `  <. A ,  y
>. ) )
3 id 19 . . 3  |-  ( x  =  A  ->  x  =  A )
42, 3eqeq12d 2310 . 2  |-  ( x  =  A  ->  (
( 1st `  <. x ,  y >. )  =  x  <->  ( 1st `  <. A ,  y >. )  =  A ) )
5 opeq2 3813 . . . 4  |-  ( y  =  B  ->  <. A , 
y >.  =  <. A ,  B >. )
65fveq2d 5545 . . 3  |-  ( y  =  B  ->  ( 1st `  <. A ,  y
>. )  =  ( 1st `  <. A ,  B >. ) )
76eqeq1d 2304 . 2  |-  ( y  =  B  ->  (
( 1st `  <. A ,  y >. )  =  A  <->  ( 1st `  <. A ,  B >. )  =  A ) )
8 vex 2804 . . 3  |-  x  e. 
_V
9 vex 2804 . . 3  |-  y  e. 
_V
108, 9op1st 6144 . 2  |-  ( 1st `  <. x ,  y
>. )  =  x
114, 7, 10vtocl2g 2860 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 1st `  <. A ,  B >. )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   <.cop 3656   ` cfv 5271   1stc1st 6136
This theorem is referenced by:  ot1stg  6150  ot2ndg  6151  1stconst  6223  curry2  6229  xpmapenlem  7044  fpwwe  8284  addpipq  8577  mulpipq  8580  ordpipq  8582  swrdval  11466  ruclem1  12525  qnumdenbi  12831  oppccofval  13635  funcf2  13758  cofuval2  13777  resfval2  13783  resf1st  13784  isnat  13837  fucco  13852  homadm  13888  setcco  13931  xpcco  13973  xpchom2  13976  xpcco2  13977  evlf2  14008  curfval  14013  curf1cl  14018  uncf1  14026  uncf2  14027  diag11  14033  diag12  14034  diag2  14035  hof2fval  14045  yonedalem21  14063  yonedalem22  14068  imasdsf1olem  17953  ovolicc1  18891  ioombl1lem3  18933  ioombl1lem4  18934  rngoablo2  21105  vcoprne  21151  brcgr  24600  fvtransport  24727  domidmor  26051  codidmor  26053  mpt2xopn0yelv  28195  mpt2xopoveq  28201  nbgraop  28274  nbgrael  28275  nbusgra  28277  dvhopvadd  31905  dvhopvsca  31914  dvhopaddN  31926  dvhopspN  31927
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-1st 6138
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