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Theorem rnsnopg 5771
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 5275 . . 3 ran {⟨𝐴, 𝐵⟩} = dom {⟨𝐴, 𝐵⟩}
2 dfdm4 5469 . . . 4 dom {⟨𝐵, 𝐴⟩} = ran {⟨𝐵, 𝐴⟩}
3 df-rn 5275 . . . 4 ran {⟨𝐵, 𝐴⟩} = dom {⟨𝐵, 𝐴⟩}
4 cnvcnvsn 5769 . . . . 5 {⟨𝐵, 𝐴⟩} = {⟨𝐴, 𝐵⟩}
54dmeqi 5478 . . . 4 dom {⟨𝐵, 𝐴⟩} = dom {⟨𝐴, 𝐵⟩}
62, 3, 53eqtri 2784 . . 3 dom {⟨𝐵, 𝐴⟩} = dom {⟨𝐴, 𝐵⟩}
71, 6eqtr4i 2783 . 2 ran {⟨𝐴, 𝐵⟩} = dom {⟨𝐵, 𝐴⟩}
8 dmsnopg 5763 . 2 (𝐴𝑉 → dom {⟨𝐵, 𝐴⟩} = {𝐵})
97, 8syl5eq 2804 1 (𝐴𝑉 → ran {⟨𝐴, 𝐵⟩} = {𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630  wcel 2137  {csn 4319  cop 4325  ccnv 5263  dom cdm 5264  ran crn 5265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pr 5053
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-rab 3057  df-v 3340  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-br 4803  df-opab 4863  df-xp 5270  df-rel 5271  df-cnv 5272  df-dm 5274  df-rn 5275
This theorem is referenced by:  rnpropg  5772  rnsnop  5774  funcnvpr  6109  funcnvtp  6110  dprdsn  18633  usgr1e  26334  1loopgredg  26605  1egrvtxdg0  26615  uspgrloopedg  26622  noextend  32123  rnsnf  39867
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