Theorem f1osn 6874

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 6607	. 2 ⊢ {⟨𝐴, 𝐵⟩} Fn {𝐴}
4	2, 1	fnsn 6607	. . 3 ⊢ {⟨𝐵, 𝐴⟩} Fn {𝐵}
5	1, 2	cnvsn 6226	. . . 4 ⊢ ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}
6	5	fneq1i 6647	. . 3 ⊢ (◡{⟨𝐴, 𝐵⟩} Fn {𝐵} ↔ {⟨𝐵, 𝐴⟩} Fn {𝐵})
7	4, 6	mpbir 230	. 2 ⊢ ◡{⟨𝐴, 𝐵⟩} Fn {𝐵}
8		dff1o4 6842	. 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 2107  Vcvv 3475  {csn 4629  ⟨cop 4635  ^◡ccnv 5676   Fn wfn 6539  –1-1-onto→wf1o 6543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551

This theorem is referenced by:  f1osng  6875  fsn  7133  ensn1  9017  ensn1OLD  9018  pssnn  9168  phplem2OLD  9218  isinf  9260  isinfOLD  9261  pssnnOLD  9265  ac6sfi  9287  marypha1lem  9428  hashf1lem1  14415  hashf1lem1OLD  14416  0ram  16953  mdet0f1o  22095  imasdsf1olem  23879  istrkg2ld  27711  axlowdimlem10  28209  subfacp1lem5  34175  poimirlem3  36491  grposnOLD  36750