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Theorem op1stg 7987
Description: Extract the first member of an ordered pair. (Contributed by NM, 19-Jul-2005.)
Assertion
Ref Expression
op1stg ((𝐴𝑉𝐵𝑊) → (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 4874 . . . 4 (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
21fveq2d 6896 . . 3 (𝑥 = 𝐴 → (1st ‘⟨𝑥, 𝑦⟩) = (1st ‘⟨𝐴, 𝑦⟩))
3 id 22 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
42, 3eqeq12d 2749 . 2 (𝑥 = 𝐴 → ((1st ‘⟨𝑥, 𝑦⟩) = 𝑥 ↔ (1st ‘⟨𝐴, 𝑦⟩) = 𝐴))
5 opeq2 4875 . . 3 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
65fveqeq2d 6900 . 2 (𝑦 = 𝐵 → ((1st ‘⟨𝐴, 𝑦⟩) = 𝐴 ↔ (1st ‘⟨𝐴, 𝐵⟩) = 𝐴))
7 vex 3479 . . 3 𝑥 ∈ V
8 vex 3479 . . 3 𝑦 ∈ V
97, 8op1st 7983 . 2 (1st ‘⟨𝑥, 𝑦⟩) = 𝑥
104, 6, 9vtocl2g 3563 1 ((𝐴𝑉𝐵𝑊) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  cop 4635  cfv 6544  1st c1st 7973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fv 6552  df-1st 7975
This theorem is referenced by:  ot1stg  7989  ot2ndg  7990  br1steqg  7997  1stconst  8086  mposn  8089  curry2  8093  opco1  8109  mpoxopn0yelv  8198  mpoxopoveq  8204  xpmapenlem  9144  1stinl  9922  1stinr  9924  fpwwe  10641  addpipq  10932  mulpipq  10935  ordpipq  10937  swrdval  14593  ruclem1  16174  qnumdenbi  16680  setsstruct  17109  oppccofval  17661  funcf2  17818  cofuval2  17837  resfval2  17843  resf1st  17844  isnat  17898  fucco  17915  homadm  17990  setcco  18033  estrcco  18081  xpcco  18135  xpchom2  18138  xpcco2  18139  evlf2  18171  curfval  18176  curf1cl  18181  uncf1  18189  uncf2  18190  diag11  18196  diag12  18197  diag2  18198  hof2fval  18208  yonedalem21  18226  yonedalem22  18231  mvmulfval  22044  imasdsf1olem  23879  ovolicc1  25033  ioombl1lem3  25077  ioombl1lem4  25078  addsqnreup  26946  addsval  27446  mulsval  27565  brcgr  28158  opvtxfv  28264  fgreu  31897  fsuppcurry2  31951  sategoelfvb  34410  prv1n  34422  fvtransport  35004  bj-inftyexpiinv  36089  bj-finsumval0  36166  poimirlem17  36505  poimirlem24  36512  poimirlem27  36515  rngoablo2  36777  dvhopvadd  39964  dvhopvsca  39973  dvhopaddN  39985  dvhopspN  39986  etransclem44  44994  ovnsubaddlem1  45286  ovnlecvr2  45326  ovolval5lem2  45369  rngccoALTV  46886  ringccoALTV  46949
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