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Theorem op1sta 6081
Description: Extract the first member of an ordered pair. (See op2nda 6084 to extract the second member, op1stb 5360 for an alternate version, and op1st 7693 for the preferred version.) (Contributed by Raph Levien, 4-Dec-2003.)
Hypotheses
Ref Expression
cnvsn.1 𝐴 ∈ V
cnvsn.2 𝐵 ∈ V
Assertion
Ref Expression
op1sta dom {⟨𝐴, 𝐵⟩} = 𝐴

Proof of Theorem op1sta
StepHypRef Expression
1 cnvsn.2 . . . 4 𝐵 ∈ V
21dmsnop 6072 . . 3 dom {⟨𝐴, 𝐵⟩} = {𝐴}
32unieqi 4846 . 2 dom {⟨𝐴, 𝐵⟩} = {𝐴}
4 cnvsn.1 . . 3 𝐴 ∈ V
54unisn 4853 . 2 {𝐴} = 𝐴
63, 5eqtri 2849 1 dom {⟨𝐴, 𝐵⟩} = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1530  wcel 2107  Vcvv 3500  {csn 4564  cop 4570   cuni 4837  dom cdm 5554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-dm 5564
This theorem is referenced by:  elxp4  7620  op1st  7693  fo1st  7705  f1stres  7709  xpassen  8605  xpdom2  8606
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