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Theorem mof0 48720
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 5305 . . . 4 ∅ ∈ V
2 eqeq2 2748 . . . . . 6 (𝑔 = ∅ → (𝑓 = 𝑔𝑓 = ∅))
32imbi2d 340 . . . . 5 (𝑔 = ∅ → ((𝑓:𝐴⟶∅ → 𝑓 = 𝑔) ↔ (𝑓:𝐴⟶∅ → 𝑓 = ∅)))
43albidv 1920 . . . 4 (𝑔 = ∅ → (∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = 𝑔) ↔ ∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = ∅)))
51, 4spcev 3605 . . 3 (∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = ∅) → ∃𝑔𝑓(𝑓:𝐴⟶∅ → 𝑓 = 𝑔))
6 f00 6788 . . . 4 (𝑓:𝐴⟶∅ ↔ (𝑓 = ∅ ∧ 𝐴 = ∅))
76simplbi 497 . . 3 (𝑓:𝐴⟶∅ → 𝑓 = ∅)
85, 7mpg 1797 . 2 𝑔𝑓(𝑓:𝐴⟶∅ → 𝑓 = 𝑔)
9 df-mo 2539 . 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 1538   = wceq 1540  wex 1779  ∃*wmo 2537  c0 4332  wf 6555
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 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430
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 2539  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5142  df-opab 5204  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-fun 6561  df-fn 6562  df-f 6563
This theorem is referenced by:  mof02  48721
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