Theorem f1osn 6656

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 6414	. 2 ⊢ {⟨𝐴, 𝐵⟩} Fn {𝐴}
4	2, 1	fnsn 6414	. . 3 ⊢ {⟨𝐵, 𝐴⟩} Fn {𝐵}
5	1, 2	cnvsn 6085	. . . 4 ⊢ ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}
6	5	fneq1i 6452	. . 3 ⊢ (◡{⟨𝐴, 𝐵⟩} Fn {𝐵} ↔ {⟨𝐵, 𝐴⟩} Fn {𝐵})
7	4, 6	mpbir 233	. 2 ⊢ ◡{⟨𝐴, 𝐵⟩} Fn {𝐵}
8		dff1o4 6625	. 2 ⊢ ({⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵} ↔ ({⟨𝐴, 𝐵⟩} Fn {𝐴} ∧ ◡{⟨𝐴, 𝐵⟩} Fn {𝐵}))
9	3, 7, 8	mpbir2an 709	1 ⊢ {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}

Syntax hints:   ∈ wcel 2114  Vcvv 3496  {csn 4569  ⟨cop 4575  ^◡ccnv 5556   Fn wfn 6352  –1-1-onto→wf1o 6356

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364

This theorem is referenced by:  f1osng  6657  fsn  6899  ensn1  8575  phplem2  8699  isinf  8733  pssnn  8738  ac6sfi  8764  marypha1lem  8899  hashf1lem1  13816  0ram  16358  mdet0f1o  21204  imasdsf1olem  22985  istrkg2ld  26248  axlowdimlem10  26739  subfacp1lem5  32433  poimirlem3  34897  grposnOLD  35162