Theorem f1osn 5503

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 5272	. 2 ⊢ {⟨𝐴, 𝐵⟩} Fn {𝐴}
4	2, 1	fnsn 5272	. . 3 ⊢ {⟨𝐵, 𝐴⟩} Fn {𝐵}
5	1, 2	cnvsn 5113	. . . 4 ⊢ ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}
6	5	fneq1i 5312	. . 3 ⊢ (◡{⟨𝐴, 𝐵⟩} Fn {𝐵} ↔ {⟨𝐵, 𝐴⟩} Fn {𝐵})
7	4, 6	mpbir 146	. 2 ⊢ ◡{⟨𝐴, 𝐵⟩} Fn {𝐵}
8		dff1o4 5471	. 2 ⊢ ({⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵} ↔ ({⟨𝐴, 𝐵⟩} Fn {𝐴} ∧ ◡{⟨𝐴, 𝐵⟩} Fn {𝐵}))
9	3, 7, 8	mpbir2an 942	1 ⊢ {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}

Syntax hints:   ∈ wcel 2148  Vcvv 2739  {csn 3594  ⟨cop 3597  ^◡ccnv 4627   Fn wfn 5213  –1-1-onto→wf1o 5217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225

This theorem is referenced by:  f1osng  5504  fsn  5690  mapsn  6692  ensn1  6798  phplem2  6855  ac6sfi  6900  fxnn0nninf  10440