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Theorem rnsnop 4829
Description: The range of a singleton of an ordered pair is the singleton of the second member. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypothesis
Ref Expression
cnvsn.1 𝐴 ∈ V
Assertion
Ref Expression
rnsnop ran {⟨𝐴, 𝐵⟩} = {𝐵}

Proof of Theorem rnsnop
StepHypRef Expression
1 cnvsn.1 . 2 𝐴 ∈ V
2 rnsnopg 4827 . 2 (𝐴 ∈ V → ran {⟨𝐴, 𝐵⟩} = {𝐵})
31, 2ax-mp 7 1 ran {⟨𝐴, 𝐵⟩} = {𝐵}
Colors of variables: wff set class
Syntax hints:   = wceq 1259  wcel 1409  Vcvv 2574  {csn 3403  cop 3406  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-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:  op2nda  4833  fpr  5373  en1  6310
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