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Theorem f1osn 5480
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 5250 . 2 {⟨𝐴, 𝐵⟩} Fn {𝐴}
42, 1fnsn 5250 . . 3 {⟨𝐵, 𝐴⟩} Fn {𝐵}
51, 2cnvsn 5091 . . . 4 {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}
65fneq1i 5290 . . 3 ({⟨𝐴, 𝐵⟩} Fn {𝐵} ↔ {⟨𝐵, 𝐴⟩} Fn {𝐵})
74, 6mpbir 145 . 2 {⟨𝐴, 𝐵⟩} Fn {𝐵}
8 dff1o4 5448 . 2 ({⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵} ↔ ({⟨𝐴, 𝐵⟩} Fn {𝐴} ∧ {⟨𝐴, 𝐵⟩} Fn {𝐵}))
93, 7, 8mpbir2an 937 1 {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}
Colors of variables: wff set class
Syntax hints:  wcel 2141  Vcvv 2730  {csn 3581  cop 3584  ccnv 4608   Fn wfn 5191  1-1-ontowf1o 5195
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-opab 4049  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203
This theorem is referenced by:  f1osng  5481  fsn  5665  mapsn  6664  ensn1  6770  phplem2  6827  ac6sfi  6872  fxnn0nninf  10381
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