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Theorem op1sta 4745
Description: Extract the first member of an ordered pair. (See op2nda 4748 to extract the second member and op1stb 4175 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 4737 . . 3 dom {⟨A, B⟩} = {A}
32unieqi 3581 . 2 dom {⟨A, B⟩} = {A}
4 cnvsn.1 . . 3 A V
54unisn 3587 . 2 {A} = A
63, 5eqtri 2057 1 dom {⟨A, B⟩} = A
Colors of variables: wff set class
Syntax hints:   = wceq 1242   wcel 1390  Vcvv 2551  {csn 3367  cop 3370   cuni 3571  dom cdm 4288
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-dm 4298
This theorem is referenced by:  op1st  5715  fo1st  5726  f1stres  5728  xpassen  6240  xpdom2  6241
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