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Theorem op1stg 6361
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 3986 . . . 4  |-  ( x  =  A  ->  <. x ,  y >.  =  <. A ,  y >. )
21fveq2d 5734 . . 3  |-  ( x  =  A  ->  ( 1st `  <. x ,  y
>. )  =  ( 1st `  <. A ,  y
>. ) )
3 id 21 . . 3  |-  ( x  =  A  ->  x  =  A )
42, 3eqeq12d 2452 . 2  |-  ( x  =  A  ->  (
( 1st `  <. x ,  y >. )  =  x  <->  ( 1st `  <. A ,  y >. )  =  A ) )
5 opeq2 3987 . . . 4  |-  ( y  =  B  ->  <. A , 
y >.  =  <. A ,  B >. )
65fveq2d 5734 . . 3  |-  ( y  =  B  ->  ( 1st `  <. A ,  y
>. )  =  ( 1st `  <. A ,  B >. ) )
76eqeq1d 2446 . 2  |-  ( y  =  B  ->  (
( 1st `  <. A ,  y >. )  =  A  <->  ( 1st `  <. A ,  B >. )  =  A ) )
8 vex 2961 . . 3  |-  x  e. 
_V
9 vex 2961 . . 3  |-  y  e. 
_V
108, 9op1st 6357 . 2  |-  ( 1st `  <. x ,  y
>. )  =  x
114, 7, 10vtocl2g 3017 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 1st `  <. A ,  B >. )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   <.cop 3819   ` cfv 5456   1stc1st 6349
This theorem is referenced by:  ot1stg  6363  ot2ndg  6364  1stconst  6437  curry2  6443  mpt2xopn0yelv  6466  mpt2xopoveq  6472  xpmapenlem  7276  fpwwe  8523  addpipq  8816  mulpipq  8819  ordpipq  8821  swrdval  11766  ruclem1  12832  qnumdenbi  13138  oppccofval  13944  funcf2  14067  cofuval2  14086  resfval2  14092  resf1st  14093  isnat  14146  fucco  14161  homadm  14197  setcco  14240  xpcco  14282  xpchom2  14285  xpcco2  14286  evlf2  14317  curfval  14322  curf1cl  14327  uncf1  14335  uncf2  14336  diag11  14342  diag12  14343  diag2  14344  hof2fval  14354  yonedalem21  14372  yonedalem22  14377  imasdsf1olem  18405  ovolicc1  19414  ioombl1lem3  19456  ioombl1lem4  19457  nbgraop  21438  rngoablo2  22012  vcoprne  22060  brcgr  25841  fvtransport  25968  dvhopvadd  31953  dvhopvsca  31962  dvhopaddN  31974  dvhopspN  31975
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-iota 5420  df-fun 5458  df-fv 5464  df-1st 6351
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