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Theorem f1sng 6090
 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 6089 . . 3 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵})
2 f1of1 6049 . . 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 4280 . . 3 (𝐵𝑊 → {𝐵} ⊆ 𝑊)
54adantl 481 . 2 ((𝐴𝑉𝐵𝑊) → {𝐵} ⊆ 𝑊)
6 f1ss 6019 . 2 (({⟨𝐴, 𝐵⟩}:{𝐴}–1-1→{𝐵} ∧ {𝐵} ⊆ 𝑊) → {⟨𝐴, 𝐵⟩}:{𝐴}–1-1𝑊)
73, 5, 6syl2anc 691 1 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩}:{𝐴}–1-1𝑊)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977   ⊆ wss 3540  {csn 4125  ⟨cop 4131  –1-1→wf1 5801  –1-1-onto→wf1o 5803 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811 This theorem is referenced by:  fsnd  6091  uspgr1e  40470  0wlkOns1  41289
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