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Theorem fnsng 6617
Description: Functionality and domain of the singleton of an ordered pair. (Contributed by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
fnsng ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} Fn {𝐴})

Proof of Theorem fnsng
StepHypRef Expression
1 funsng 6616 . 2 ((𝐴𝑉𝐵𝑊) → Fun {⟨𝐴, 𝐵⟩})
2 dmsnopg 6232 . . 3 (𝐵𝑊 → dom {⟨𝐴, 𝐵⟩} = {𝐴})
32adantl 481 . 2 ((𝐴𝑉𝐵𝑊) → dom {⟨𝐴, 𝐵⟩} = {𝐴})
4 df-fn 6563 . 2 ({⟨𝐴, 𝐵⟩} Fn {𝐴} ↔ (Fun {⟨𝐴, 𝐵⟩} ∧ dom {⟨𝐴, 𝐵⟩} = {𝐴}))
51, 3, 4sylanbrc 583 1 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} Fn {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  {csn 4625  cop 4631  dom cdm 5684  Fun wfun 6554   Fn wfn 6555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-mo 2539  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-fun 6562  df-fn 6563
This theorem is referenced by:  fnsn  6623  fnunop  6683  fnsnbg  7185  fvsnun2  7204  fsnunfv  7208  mat1dimscm  22482  m1detdiag  22604  noextenddif  27714  noextendlt  27715  noextendgt  27716  actfunsnf1o  34620  actfunsnrndisj  34621  breprexplema  34646  sticksstones11  42158  metakunt19  42225
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