Theorem f1osn 5472

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

Step	Hyp	Ref	Expression
1		f1osn.1	. . 3 ⊢ 𝐴 ∈ V
2		f1osn.2	. . 3 ⊢ 𝐵 ∈ V
3	1, 2	fnsn 5242	. 2 ⊢ {⟨𝐴, 𝐵⟩} Fn {𝐴}
4	2, 1	fnsn 5242	. . 3 ⊢ {⟨𝐵, 𝐴⟩} Fn {𝐵}
5	1, 2	cnvsn 5086	. . . 4 ⊢ ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}
6	5	fneq1i 5282	. . 3 ⊢ (◡{⟨𝐴, 𝐵⟩} Fn {𝐵} ↔ {⟨𝐵, 𝐴⟩} Fn {𝐵})
7	4, 6	mpbir 145	. 2 ⊢ ◡{⟨𝐴, 𝐵⟩} Fn {𝐵}
8		dff1o4 5440	. 2 ⊢ ({⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵} ↔ ({⟨𝐴, 𝐵⟩} Fn {𝐴} ∧ ◡{⟨𝐴, 𝐵⟩} Fn {𝐵}))
9	3, 7, 8	mpbir2an 932	1 ⊢ {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}

Syntax hints:   ∈ wcel 2136  Vcvv 2726  {csn 3576  ⟨cop 3579  ^◡ccnv 4603   Fn wfn 5183  –1-1-onto→wf1o 5187

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195

This theorem is referenced by:  f1osng  5473  fsn  5657  mapsn  6656  ensn1  6762  phplem2  6819  ac6sfi  6864  fxnn0nninf  10373