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Theorem rnsnopg 5573
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 5085 . . 3 ran {⟨𝐴, 𝐵⟩} = dom {⟨𝐴, 𝐵⟩}
2 dfdm4 5276 . . . 4 dom {⟨𝐵, 𝐴⟩} = ran {⟨𝐵, 𝐴⟩}
3 df-rn 5085 . . . 4 ran {⟨𝐵, 𝐴⟩} = dom {⟨𝐵, 𝐴⟩}
4 cnvcnvsn 5571 . . . . 5 {⟨𝐵, 𝐴⟩} = {⟨𝐴, 𝐵⟩}
54dmeqi 5285 . . . 4 dom {⟨𝐵, 𝐴⟩} = dom {⟨𝐴, 𝐵⟩}
62, 3, 53eqtri 2647 . . 3 dom {⟨𝐵, 𝐴⟩} = dom {⟨𝐴, 𝐵⟩}
71, 6eqtr4i 2646 . 2 ran {⟨𝐴, 𝐵⟩} = dom {⟨𝐵, 𝐴⟩}
8 dmsnopg 5565 . 2 (𝐴𝑉 → dom {⟨𝐵, 𝐴⟩} = {𝐵})
97, 8syl5eq 2667 1 (𝐴𝑉 → ran {⟨𝐴, 𝐵⟩} = {𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  {csn 4148  cop 4154  ccnv 5073  dom cdm 5074  ran crn 5075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-cnv 5082  df-dm 5084  df-rn 5085
This theorem is referenced by:  rnpropg  5574  rnsnop  5575  funcnvpr  5908  funcnvtp  5909  dprdsn  18356  usgr1e  26030  1loopgredg  26283  1egrvtxdg0  26293  uspgrloopedg  26300  noextend  31521  rnsnf  38841
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