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Theorem f102g 48682
Description: A function that maps the empty set to a class is injective. (Contributed by Zhi Wang, 1-Oct-2024.)
Assertion
Ref Expression
f102g ((𝐴 = ∅ ∧ 𝐹:𝐴𝐵) → 𝐹:𝐴1-1𝐵)

Proof of Theorem f102g
StepHypRef Expression
1 feq2 6718 . . . 4 (𝐴 = ∅ → (𝐹:𝐴𝐵𝐹:∅⟶𝐵))
21biimpa 476 . . 3 ((𝐴 = ∅ ∧ 𝐹:𝐴𝐵) → 𝐹:∅⟶𝐵)
3 f0bi 6792 . . . 4 (𝐹:∅⟶𝐵𝐹 = ∅)
4 f10 6882 . . . . 5 ∅:∅–1-1𝐵
5 f1eq1 6800 . . . . 5 (𝐹 = ∅ → (𝐹:∅–1-1𝐵 ↔ ∅:∅–1-1𝐵))
64, 5mpbiri 258 . . . 4 (𝐹 = ∅ → 𝐹:∅–1-1𝐵)
73, 6sylbi 217 . . 3 (𝐹:∅⟶𝐵𝐹:∅–1-1𝐵)
82, 7syl 17 . 2 ((𝐴 = ∅ ∧ 𝐹:𝐴𝐵) → 𝐹:∅–1-1𝐵)
9 f1eq2 6801 . . 3 (𝐴 = ∅ → (𝐹:𝐴1-1𝐵𝐹:∅–1-1𝐵))
109adantr 480 . 2 ((𝐴 = ∅ ∧ 𝐹:𝐴𝐵) → (𝐹:𝐴1-1𝐵𝐹:∅–1-1𝐵))
118, 10mpbird 257 1 ((𝐴 = ∅ ∧ 𝐹:𝐴𝐵) → 𝐹:𝐴1-1𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  c0 4339  wf 6559  1-1wf1 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568
This theorem is referenced by:  f1mo  48683
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