Theorem rnsnm 5093

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 5092	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ dom {𝐴})
2		dmmrnm 4844	. 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 1492   ∈ wcel 2148  Vcvv 2737  {csn 3592   × cxp 4623  dom cdm 4625  ran crn 4626

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4003  df-opab 4064  df-xp 4631  df-cnv 4633  df-dm 4635  df-rn 4636

This theorem is referenced by: (None)