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Theorem op1sta 4830
Description: Extract the first member of an ordered pair. (See op2nda 4833 to extract the second member and op1stb 4237 for an alternate 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 4822 . . 3 dom {⟨𝐴, 𝐵⟩} = {𝐴}
32unieqi 3618 . 2 dom {⟨𝐴, 𝐵⟩} = {𝐴}
4 cnvsn.1 . . 3 𝐴 ∈ V
54unisn 3624 . 2 {𝐴} = 𝐴
63, 5eqtri 2076 1 dom {⟨𝐴, 𝐵⟩} = 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1259  wcel 1409  Vcvv 2574  {csn 3403  cop 3406   cuni 3608  dom cdm 4373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-dm 4383
This theorem is referenced by:  op1st  5801  fo1st  5812  f1stres  5814  xpassen  6335  xpdom2  6336
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