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Theorem f102g 48761
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 6717 . . . 4 (𝐴 = ∅ → (𝐹:𝐴𝐵𝐹:∅⟶𝐵))
21biimpa 476 . . 3 ((𝐴 = ∅ ∧ 𝐹:𝐴𝐵) → 𝐹:∅⟶𝐵)
3 f0bi 6791 . . . 4 (𝐹:∅⟶𝐵𝐹 = ∅)
4 f10 6881 . . . . 5 ∅:∅–1-1𝐵
5 f1eq1 6799 . . . . 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 6800 . . 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 1540  c0 4333  wf 6557  1-1wf1 6558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566
This theorem is referenced by:  f1mo  48762
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