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Theorem funsn 5096
Description: A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 12-Aug-1994.)
Hypotheses
Ref Expression
funsn.1 𝐴 ∈ V
funsn.2 𝐵 ∈ V
Assertion
Ref Expression
funsn Fun {⟨𝐴, 𝐵⟩}

Proof of Theorem funsn
StepHypRef Expression
1 funsn.1 . 2 𝐴 ∈ V
2 funsn.2 . 2 𝐵 ∈ V
3 funsng 5094 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → Fun {⟨𝐴, 𝐵⟩})
41, 2, 3mp2an 418 1 Fun {⟨𝐴, 𝐵⟩}
Colors of variables: wff set class
Syntax hints:  wcel 1445  Vcvv 2633  {csn 3466  cop 3469  Fun wfun 5043
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-br 3868  df-opab 3922  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-fun 5051
This theorem is referenced by:  funtp  5101  fun0  5106  fvsn  5531
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