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Theorem rnsnopg 4827
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, 30-Apr-2015.)
Assertion
Ref Expression
rnsnopg (𝐴𝑉 → ran {⟨𝐴, 𝐵⟩} = {𝐵})

Proof of Theorem rnsnopg
StepHypRef Expression
1 df-rn 4384 . . 3 ran {⟨𝐴, 𝐵⟩} = dom {⟨𝐴, 𝐵⟩}
2 dfdm4 4555 . . . 4 dom {⟨𝐵, 𝐴⟩} = ran {⟨𝐵, 𝐴⟩}
3 df-rn 4384 . . . 4 ran {⟨𝐵, 𝐴⟩} = dom {⟨𝐵, 𝐴⟩}
4 cnvcnvsn 4825 . . . . 5 {⟨𝐵, 𝐴⟩} = {⟨𝐴, 𝐵⟩}
54dmeqi 4564 . . . 4 dom {⟨𝐵, 𝐴⟩} = dom {⟨𝐴, 𝐵⟩}
62, 3, 53eqtri 2080 . . 3 dom {⟨𝐵, 𝐴⟩} = dom {⟨𝐴, 𝐵⟩}
71, 6eqtr4i 2079 . 2 ran {⟨𝐴, 𝐵⟩} = dom {⟨𝐵, 𝐴⟩}
8 dmsnopg 4820 . 2 (𝐴𝑉 → dom {⟨𝐵, 𝐴⟩} = {𝐵})
97, 8syl5eq 2100 1 (𝐴𝑉 → ran {⟨𝐴, 𝐵⟩} = {𝐵})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  wcel 1409  {csn 3403  cop 3406  ccnv 4372  dom cdm 4373  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:  rnpropg  4828  rnsnop  4829  fprg  5374
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