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Theorem mof0 48596
Description: There is at most one function into the empty set. (Contributed by Zhi Wang, 19-Sep-2024.)
Assertion
Ref Expression
mof0 ∃*𝑓 𝑓:𝐴⟶∅

Proof of Theorem mof0
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 0ex 5309 . . . 4 ∅ ∈ V
2 eqeq2 2745 . . . . . 6 (𝑔 = ∅ → (𝑓 = 𝑔𝑓 = ∅))
32imbi2d 340 . . . . 5 (𝑔 = ∅ → ((𝑓:𝐴⟶∅ → 𝑓 = 𝑔) ↔ (𝑓:𝐴⟶∅ → 𝑓 = ∅)))
43albidv 1916 . . . 4 (𝑔 = ∅ → (∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = 𝑔) ↔ ∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = ∅)))
51, 4spcev 3606 . . 3 (∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = ∅) → ∃𝑔𝑓(𝑓:𝐴⟶∅ → 𝑓 = 𝑔))
6 f00 6786 . . . 4 (𝑓:𝐴⟶∅ ↔ (𝑓 = ∅ ∧ 𝐴 = ∅))
76simplbi 497 . . 3 (𝑓:𝐴⟶∅ → 𝑓 = ∅)
85, 7mpg 1792 . 2 𝑔𝑓(𝑓:𝐴⟶∅ → 𝑓 = 𝑔)
9 df-mo 2536 . 2 (∃*𝑓 𝑓:𝐴⟶∅ ↔ ∃𝑔𝑓(𝑓:𝐴⟶∅ → 𝑓 = 𝑔))
108, 9mpbir 231 1 ∃*𝑓 𝑓:𝐴⟶∅
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1533   = wceq 1535  wex 1774  ∃*wmo 2534  c0 4339  wf 6555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704  ax-sep 5301  ax-nul 5308  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2536  df-clab 2711  df-cleq 2725  df-clel 2812  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  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 5151  df-opab 5213  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-fun 6561  df-fn 6562  df-f 6563
This theorem is referenced by:  mof02  48597
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