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Theorem mof0 45742
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 5185 . . . 4 ∅ ∈ V
2 eqeq2 2751 . . . . . 6 (𝑔 = ∅ → (𝑓 = 𝑔𝑓 = ∅))
32imbi2d 344 . . . . 5 (𝑔 = ∅ → ((𝑓:𝐴⟶∅ → 𝑓 = 𝑔) ↔ (𝑓:𝐴⟶∅ → 𝑓 = ∅)))
43albidv 1927 . . . 4 (𝑔 = ∅ → (∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = 𝑔) ↔ ∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = ∅)))
51, 4spcev 3513 . . 3 (∀𝑓(𝑓:𝐴⟶∅ → 𝑓 = ∅) → ∃𝑔𝑓(𝑓:𝐴⟶∅ → 𝑓 = 𝑔))
6 f00 6570 . . . 4 (𝑓:𝐴⟶∅ ↔ (𝑓 = ∅ ∧ 𝐴 = ∅))
76simplbi 501 . . 3 (𝑓:𝐴⟶∅ → 𝑓 = ∅)
85, 7mpg 1804 . 2 𝑔𝑓(𝑓:𝐴⟶∅ → 𝑓 = 𝑔)
9 df-mo 2541 . 2 (∃*𝑓 𝑓:𝐴⟶∅ ↔ ∃𝑔𝑓(𝑓:𝐴⟶∅ → 𝑓 = 𝑔))
108, 9mpbir 234 1 ∃*𝑓 𝑓:𝐴⟶∅
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1540   = wceq 1542  wex 1786  ∃*wmo 2539  c0 4221  wf 6345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ral 3059  df-rex 3060  df-v 3402  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-br 5041  df-opab 5103  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-fun 6351  df-fn 6352  df-f 6353
This theorem is referenced by:  mof02  45743
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