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Theorem rnsnm 4815
 Description: The range of a singleton is inhabited iff the singleton argument is an ordered pair. (Contributed by Jim Kingdon, 15-Dec-2018.)
Assertion
Ref Expression
rnsnm (𝐴 ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ ran {𝐴})
Distinct variable group:   𝑥,𝐴

Proof of Theorem rnsnm
StepHypRef Expression
1 dmsnm 4814 . 2 (𝐴 ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ dom {𝐴})
2 dmmrnm 4582 . 2 (∃𝑥 𝑥 ∈ dom {𝐴} ↔ ∃𝑥 𝑥 ∈ ran {𝐴})
31, 2bitri 177 1 (𝐴 ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ ran {𝐴})
 Colors of variables: wff set class Syntax hints:   ↔ wb 102  ∃wex 1397   ∈ wcel 1409  Vcvv 2574  {csn 3403   × cxp 4371  dom cdm 4373  ran crn 4374 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379  df-cnv 4381  df-dm 4383  df-rn 4384 This theorem is referenced by: (None)
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