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Theorem rnsnopg 6213
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 5680 . . 3 ran {⟨𝐴, 𝐵⟩} = dom {⟨𝐴, 𝐵⟩}
2 dfdm4 5888 . . . 4 dom {⟨𝐵, 𝐴⟩} = ran {⟨𝐵, 𝐴⟩}
3 df-rn 5680 . . . 4 ran {⟨𝐵, 𝐴⟩} = dom {⟨𝐵, 𝐴⟩}
4 cnvcnvsn 6211 . . . . 5 {⟨𝐵, 𝐴⟩} = {⟨𝐴, 𝐵⟩}
54dmeqi 5897 . . . 4 dom {⟨𝐵, 𝐴⟩} = dom {⟨𝐴, 𝐵⟩}
62, 3, 53eqtri 2758 . . 3 dom {⟨𝐵, 𝐴⟩} = dom {⟨𝐴, 𝐵⟩}
71, 6eqtr4i 2757 . 2 ran {⟨𝐴, 𝐵⟩} = dom {⟨𝐵, 𝐴⟩}
8 dmsnopg 6205 . 2 (𝐴𝑉 → dom {⟨𝐵, 𝐴⟩} = {𝐵})
97, 8eqtrid 2778 1 (𝐴𝑉 → ran {⟨𝐴, 𝐵⟩} = {𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  {csn 4623  cop 4629  ccnv 5668  dom cdm 5669  ran crn 5670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680
This theorem is referenced by:  rnpropg  6214  rnsnop  6216  funcnvpr  6603  funcnvtp  6604  dprdsn  19956  noextend  27550  usgr1e  29006  1loopgredg  29263  1egrvtxdg0  29273  uspgrloopedg  29280  cosnopne  32421  rnsnf  44436
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