Theorem rnsnm 5148

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

Step	Hyp	Ref	Expression
1		dmsnm 5147	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ dom {𝐴})
2		dmmrnm 4896	. 2 ⊢ (∃𝑥 𝑥 ∈ dom {𝐴} ↔ ∃𝑥 𝑥 ∈ ran {𝐴})
3	1, 2	bitri 184	1 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ ran {𝐴})

Syntax hints:   ↔ wb 105  ∃wex 1514   ∈ wcel 2175  Vcvv 2771  {csn 3632   × cxp 4672  dom cdm 4674  ran crn 4675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-xp 4680  df-cnv 4682  df-dm 4684  df-rn 4685

This theorem is referenced by: (None)