MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1sng Structured version   Visualization version   GIF version

Theorem f1sng 6629
Description: A singleton of an ordered pair is a one-to-one function. (Contributed by AV, 17-Apr-2021.)
Assertion
Ref Expression
f1sng ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩}:{𝐴}–1-1𝑊)

Proof of Theorem f1sng
StepHypRef Expression
1 f1osng 6628 . . 3 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵})
2 f1of1 6587 . . 3 ({⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵} → {⟨𝐴, 𝐵⟩}:{𝐴}–1-1→{𝐵})
31, 2syl 17 . 2 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩}:{𝐴}–1-1→{𝐵})
4 snssi 4714 . . 3 (𝐵𝑊 → {𝐵} ⊆ 𝑊)
54adantl 485 . 2 ((𝐴𝑉𝐵𝑊) → {𝐵} ⊆ 𝑊)
6 f1ss 6553 . 2 (({⟨𝐴, 𝐵⟩}:{𝐴}–1-1→{𝐵} ∧ {𝐵} ⊆ 𝑊) → {⟨𝐴, 𝐵⟩}:{𝐴}–1-1𝑊)
73, 5, 6syl2anc 587 1 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩}:{𝐴}–1-1𝑊)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2115  wss 3910  {csn 4540  cop 4546  1-1wf1 6325  1-1-ontowf1o 6327
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-br 5040  df-opab 5102  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335
This theorem is referenced by:  fsnd  6630  uspgr1e  27012  0wlkons1  27884
  Copyright terms: Public domain W3C validator