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Theorem fnsng 4890
Description: Functionality and domain of the singleton of an ordered pair. (Contributed by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
fnsng ((A 𝑉 B 𝑊) → {⟨A, B⟩} Fn {A})

Proof of Theorem fnsng
StepHypRef Expression
1 funsng 4889 . 2 ((A 𝑉 B 𝑊) → Fun {⟨A, B⟩})
2 dmsnopg 4735 . . 3 (B 𝑊 → dom {⟨A, B⟩} = {A})
32adantl 262 . 2 ((A 𝑉 B 𝑊) → dom {⟨A, B⟩} = {A})
4 df-fn 4848 . 2 ({⟨A, B⟩} Fn {A} ↔ (Fun {⟨A, B⟩} dom {⟨A, B⟩} = {A}))
51, 3, 4sylanbrc 394 1 ((A 𝑉 B 𝑊) → {⟨A, B⟩} Fn {A})
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  {csn 3367  cop 3370  dom cdm 4288  Fun wfun 4839   Fn wfn 4840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-fun 4847  df-fn 4848
This theorem is referenced by:  fnsn  4896  fnunsn  4949  fsnunfv  5306  tfr0  5878
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