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Theorem op2nda 4833
Description: Extract the second member of an ordered pair. (See op1sta 4830 to extract the first member and op2ndb 4832 for an alternate version.) (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Hypotheses
Ref Expression
cnvsn.1 𝐴 ∈ V
cnvsn.2 𝐵 ∈ V
Assertion
Ref Expression
op2nda ran {⟨𝐴, 𝐵⟩} = 𝐵

Proof of Theorem op2nda
StepHypRef Expression
1 cnvsn.1 . . . 4 𝐴 ∈ V
21rnsnop 4829 . . 3 ran {⟨𝐴, 𝐵⟩} = {𝐵}
32unieqi 3618 . 2 ran {⟨𝐴, 𝐵⟩} = {𝐵}
4 cnvsn.2 . . 3 𝐵 ∈ V
54unisn 3624 . 2 {𝐵} = 𝐵
63, 5eqtri 2076 1 ran {⟨𝐴, 𝐵⟩} = 𝐵
Colors of variables: wff set class
Syntax hints:   = wceq 1259  wcel 1409  Vcvv 2574  {csn 3403  cop 3406   cuni 3608  ran crn 4374
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-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  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-opab 3847  df-xp 4379  df-rel 4380  df-cnv 4381  df-dm 4383  df-rn 4384
This theorem is referenced by:  elxp4  4836  elxp5  4837  op2nd  5802  fo2nd  5813  f2ndres  5815  xpassen  6335  xpdom2  6336
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