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Theorem mof0 47716
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 5298 . . . 4 ∅ ∈ V
2 eqeq2 2736 . . . . . 6 (𝑔 = ∅ → (𝑓 = 𝑔𝑓 = ∅))
32imbi2d 340 . . . . 5 (𝑔 = ∅ → ((𝑓:𝐴⟶∅ → 𝑓 = 𝑔) ↔ (𝑓:𝐴⟶∅ → 𝑓 = ∅)))
43albidv 1915 . . . 4 (𝑔 = ∅ → (∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = 𝑔) ↔ ∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = ∅)))
51, 4spcev 3588 . . 3 (∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = ∅) → ∃𝑔𝑓(𝑓:𝐴⟶∅ → 𝑓 = 𝑔))
6 f00 6764 . . . 4 (𝑓:𝐴⟶∅ ↔ (𝑓 = ∅ ∧ 𝐴 = ∅))
76simplbi 497 . . 3 (𝑓:𝐴⟶∅ → 𝑓 = ∅)
85, 7mpg 1791 . 2 𝑔𝑓(𝑓:𝐴⟶∅ → 𝑓 = 𝑔)
9 df-mo 2526 . 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 1531   = wceq 1533  wex 1773  ∃*wmo 2524  c0 4315  wf 6530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-fun 6536  df-fn 6537  df-f 6538
This theorem is referenced by:  mof02  47717
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