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Theorem mof0 46165
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 5231 . . . 4 ∅ ∈ V
2 eqeq2 2750 . . . . . 6 (𝑔 = ∅ → (𝑓 = 𝑔𝑓 = ∅))
32imbi2d 341 . . . . 5 (𝑔 = ∅ → ((𝑓:𝐴⟶∅ → 𝑓 = 𝑔) ↔ (𝑓:𝐴⟶∅ → 𝑓 = ∅)))
43albidv 1923 . . . 4 (𝑔 = ∅ → (∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = 𝑔) ↔ ∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = ∅)))
51, 4spcev 3545 . . 3 (∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = ∅) → ∃𝑔𝑓(𝑓:𝐴⟶∅ → 𝑓 = 𝑔))
6 f00 6656 . . . 4 (𝑓:𝐴⟶∅ ↔ (𝑓 = ∅ ∧ 𝐴 = ∅))
76simplbi 498 . . 3 (𝑓:𝐴⟶∅ → 𝑓 = ∅)
85, 7mpg 1800 . 2 𝑔𝑓(𝑓:𝐴⟶∅ → 𝑓 = 𝑔)
9 df-mo 2540 . 2 (∃*𝑓 𝑓:𝐴⟶∅ ↔ ∃𝑔𝑓(𝑓:𝐴⟶∅ → 𝑓 = 𝑔))
108, 9mpbir 230 1 ∃*𝑓 𝑓:𝐴⟶∅
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1537   = wceq 1539  wex 1782  ∃*wmo 2538  c0 4256  wf 6429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-fun 6435  df-fn 6436  df-f 6437
This theorem is referenced by:  mof02  46166
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