Theorem f1osn 6646

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 6398	. 2 ⊢ {⟨𝐴, 𝐵⟩} Fn {𝐴}
4	2, 1	fnsn 6398	. . 3 ⊢ {⟨𝐵, 𝐴⟩} Fn {𝐵}
5	1, 2	cnvsn 6060	. . . 4 ⊢ ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}
6	5	fneq1i 6436	. . 3 ⊢ (◡{⟨𝐴, 𝐵⟩} Fn {𝐵} ↔ {⟨𝐵, 𝐴⟩} Fn {𝐵})
7	4, 6	mpbir 234	. 2 ⊢ ◡{⟨𝐴, 𝐵⟩} Fn {𝐵}
8		dff1o4 6615	. 2 ⊢ ({⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵} ↔ ({⟨𝐴, 𝐵⟩} Fn {𝐴} ∧ ◡{⟨𝐴, 𝐵⟩} Fn {𝐵}))
9	3, 7, 8	mpbir2an 710	1 ⊢ {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}

Syntax hints:   ∈ wcel 2111  Vcvv 3409  {csn 4525  ⟨cop 4531  ^◡ccnv 5527   Fn wfn 6335  –1-1-onto→wf1o 6339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-br 5037  df-opab 5099  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347

This theorem is referenced by:  f1osng  6647  fsn  6894  ensn1  8605  phplem2  8732  pssnn  8751  isinf  8782  pssnnOLD  8787  ac6sfi  8808  marypha1lem  8943  hashf1lem1  13877  hashf1lem1OLD  13878  0ram  16425  mdet0f1o  21307  imasdsf1olem  23089  istrkg2ld  26367  axlowdimlem10  26858  subfacp1lem5  32675  poimirlem3  35375  grposnOLD  35635