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Theorem f1osn 5407
Description: A singleton of an ordered pair is one-to-one onto function. (Contributed by NM, 18-May-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Hypotheses
Ref Expression
f1osn.1 𝐴 ∈ V
f1osn.2 𝐵 ∈ V
Assertion
Ref Expression
f1osn {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}

Proof of Theorem f1osn
StepHypRef Expression
1 f1osn.1 . . 3 𝐴 ∈ V
2 f1osn.2 . . 3 𝐵 ∈ V
31, 2fnsn 5177 . 2 {⟨𝐴, 𝐵⟩} Fn {𝐴}
42, 1fnsn 5177 . . 3 {⟨𝐵, 𝐴⟩} Fn {𝐵}
51, 2cnvsn 5021 . . . 4 {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}
65fneq1i 5217 . . 3 ({⟨𝐴, 𝐵⟩} Fn {𝐵} ↔ {⟨𝐵, 𝐴⟩} Fn {𝐵})
74, 6mpbir 145 . 2 {⟨𝐴, 𝐵⟩} Fn {𝐵}
8 dff1o4 5375 . 2 ({⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵} ↔ ({⟨𝐴, 𝐵⟩} Fn {𝐴} ∧ {⟨𝐴, 𝐵⟩} Fn {𝐵}))
93, 7, 8mpbir2an 926 1 {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}
Colors of variables: wff set class
Syntax hints:  wcel 1480  Vcvv 2686  {csn 3527  cop 3530  ccnv 4538   Fn wfn 5118  1-1-ontowf1o 5122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130
This theorem is referenced by:  f1osng  5408  fsn  5592  mapsn  6584  ensn1  6690  phplem2  6747  ac6sfi  6792  fxnn0nninf  10218
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