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Theorem rnsnopg 5755
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 5261 . . 3 ran {⟨𝐴, 𝐵⟩} = dom {⟨𝐴, 𝐵⟩}
2 dfdm4 5453 . . . 4 dom {⟨𝐵, 𝐴⟩} = ran {⟨𝐵, 𝐴⟩}
3 df-rn 5261 . . . 4 ran {⟨𝐵, 𝐴⟩} = dom {⟨𝐵, 𝐴⟩}
4 cnvcnvsn 5753 . . . . 5 {⟨𝐵, 𝐴⟩} = {⟨𝐴, 𝐵⟩}
54dmeqi 5462 . . . 4 dom {⟨𝐵, 𝐴⟩} = dom {⟨𝐴, 𝐵⟩}
62, 3, 53eqtri 2797 . . 3 dom {⟨𝐵, 𝐴⟩} = dom {⟨𝐴, 𝐵⟩}
71, 6eqtr4i 2796 . 2 ran {⟨𝐴, 𝐵⟩} = dom {⟨𝐵, 𝐴⟩}
8 dmsnopg 5747 . 2 (𝐴𝑉 → dom {⟨𝐵, 𝐴⟩} = {𝐵})
97, 8syl5eq 2817 1 (𝐴𝑉 → ran {⟨𝐴, 𝐵⟩} = {𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  {csn 4317  cop 4323  ccnv 5249  dom cdm 5250  ran crn 5251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-xp 5256  df-rel 5257  df-cnv 5258  df-dm 5260  df-rn 5261
This theorem is referenced by:  rnpropg  5756  rnsnop  5758  funcnvpr  6090  funcnvtp  6091  dprdsn  18643  usgr1e  26360  1loopgredg  26632  1egrvtxdg0  26642  uspgrloopedg  26649  noextend  32156  rnsnf  39889
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