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Theorem f1osn 5109
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 4896 . 2 {⟨A, B⟩} Fn {A}
42, 1fnsn 4896 . . 3 {⟨B, A⟩} Fn {B}
51, 2cnvsn 4746 . . . 4 {⟨A, B⟩} = {⟨B, A⟩}
65fneq1i 4936 . . 3 ({⟨A, B⟩} Fn {B} ↔ {⟨B, A⟩} Fn {B})
74, 6mpbir 134 . 2 {⟨A, B⟩} Fn {B}
8 dff1o4 5077 . 2 ({⟨A, B⟩}:{A}–1-1-onto→{B} ↔ ({⟨A, B⟩} Fn {A} {⟨A, B⟩} Fn {B}))
93, 7, 8mpbir2an 848 1 {⟨A, B⟩}:{A}–1-1-onto→{B}
Colors of variables: wff set class
Syntax hints:   wcel 1390  Vcvv 2551  {csn 3367  cop 3370  ccnv 4287   Fn wfn 4840  1-1-ontowf1o 4844
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-bnd 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-rn 4299  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852
This theorem is referenced by:  f1osng  5110  fsn  5278  ensn1  6212
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