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Theorem op1stg 6132
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 3796 . . . 4  |-  ( x  =  A  ->  <. x ,  y >.  =  <. A ,  y >. )
21fveq2d 5529 . . 3  |-  ( x  =  A  ->  ( 1st `  <. x ,  y
>. )  =  ( 1st `  <. A ,  y
>. ) )
3 id 19 . . 3  |-  ( x  =  A  ->  x  =  A )
42, 3eqeq12d 2297 . 2  |-  ( x  =  A  ->  (
( 1st `  <. x ,  y >. )  =  x  <->  ( 1st `  <. A ,  y >. )  =  A ) )
5 opeq2 3797 . . . 4  |-  ( y  =  B  ->  <. A , 
y >.  =  <. A ,  B >. )
65fveq2d 5529 . . 3  |-  ( y  =  B  ->  ( 1st `  <. A ,  y
>. )  =  ( 1st `  <. A ,  B >. ) )
76eqeq1d 2291 . 2  |-  ( y  =  B  ->  (
( 1st `  <. A ,  y >. )  =  A  <->  ( 1st `  <. A ,  B >. )  =  A ) )
8 vex 2791 . . 3  |-  x  e. 
_V
9 vex 2791 . . 3  |-  y  e. 
_V
108, 9op1st 6128 . 2  |-  ( 1st `  <. x ,  y
>. )  =  x
114, 7, 10vtocl2g 2847 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 1st `  <. A ,  B >. )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   <.cop 3643   ` cfv 5255   1stc1st 6120
This theorem is referenced by:  ot1stg  6134  ot2ndg  6135  1stconst  6207  curry2  6213  xpmapenlem  7028  fpwwe  8268  addpipq  8561  mulpipq  8564  ordpipq  8566  swrdval  11450  ruclem1  12509  qnumdenbi  12815  oppccofval  13619  funcf2  13742  cofuval2  13761  resfval2  13767  resf1st  13768  isnat  13821  fucco  13836  homadm  13872  setcco  13915  xpcco  13957  xpchom2  13960  xpcco2  13961  evlf2  13992  curfval  13997  curf1cl  14002  uncf1  14010  uncf2  14011  diag11  14017  diag12  14018  diag2  14019  hof2fval  14029  yonedalem21  14047  yonedalem22  14052  imasdsf1olem  17937  ovolicc1  18875  ioombl1lem3  18917  ioombl1lem4  18918  rngoablo2  21089  vcoprne  21135  brcgr  24528  fvtransport  24655  domidmor  25948  codidmor  25950  mpt2xopn0yelv  28079  mpt2xopoveq  28085  nbgraop  28140  nbgrael  28141  nbusgra  28143  dvhopvadd  31283  dvhopvsca  31292  dvhopaddN  31304  dvhopspN  31305
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fv 5263  df-1st 6122
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