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Theorem op1sta 6178
Description: Extract the first member of an ordered pair. (See op2nda 6181 to extract the second member, op1stb 5429 for an alternate version, and op1st 7930 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 6169 . . 3 dom {⟨𝐴, 𝐵⟩} = {𝐴}
32unieqi 4879 . 2 dom {⟨𝐴, 𝐵⟩} = {𝐴}
4 cnvsn.1 . . 3 𝐴 ∈ V
54unisn 4888 . 2 {𝐴} = 𝐴
63, 5eqtri 2765 1 dom {⟨𝐴, 𝐵⟩} = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2107  Vcvv 3446  {csn 4587  cop 4593   cuni 4866  dom cdm 5634
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-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
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-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-dm 5644
This theorem is referenced by:  elxp4  7860  op1st  7930  fo1st  7942  f1stres  7946  xpassen  9011  xpdom2  9012
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