Theorem rnsnm 5168

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 5167	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ dom {𝐴})
2		dmmrnm 4916	. 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 1516   ∈ wcel 2178  Vcvv 2776  {csn 3643   × cxp 4691  dom cdm 4693  ran crn 4694

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-xp 4699  df-cnv 4701  df-dm 4703  df-rn 4704

This theorem is referenced by: (None)