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Theorem rnsnopg 4742
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 (A 𝑉 → ran {⟨A, B⟩} = {B})

Proof of Theorem rnsnopg
StepHypRef Expression
1 df-rn 4299 . . 3 ran {⟨A, B⟩} = dom {⟨A, B⟩}
2 dfdm4 4470 . . . 4 dom {⟨B, A⟩} = ran {⟨B, A⟩}
3 df-rn 4299 . . . 4 ran {⟨B, A⟩} = dom {⟨B, A⟩}
4 cnvcnvsn 4740 . . . . 5 {⟨B, A⟩} = {⟨A, B⟩}
54dmeqi 4479 . . . 4 dom {⟨B, A⟩} = dom {⟨A, B⟩}
62, 3, 53eqtri 2061 . . 3 dom {⟨B, A⟩} = dom {⟨A, B⟩}
71, 6eqtr4i 2060 . 2 ran {⟨A, B⟩} = dom {⟨B, A⟩}
8 dmsnopg 4735 . 2 (A 𝑉 → dom {⟨B, A⟩} = {B})
97, 8syl5eq 2081 1 (A 𝑉 → ran {⟨A, B⟩} = {B})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390  {csn 3367  cop 3370  ccnv 4287  dom cdm 4288  ran crn 4289
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299
This theorem is referenced by:  rnpropg  4743  rnsnop  4744  fprg  5289
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