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Theorem mof02 46166
Description: A variant of mof0 46165. (Contributed by Zhi Wang, 20-Sep-2024.)
Assertion
Ref Expression
mof02 (𝐵 = ∅ → ∃*𝑓 𝑓:𝐴𝐵)
Distinct variable group:   𝐵,𝑓
Allowed substitution hint:   𝐴(𝑓)

Proof of Theorem mof02
StepHypRef Expression
1 mof0 46165 . 2 ∃*𝑓 𝑓:𝐴⟶∅
2 feq3 6583 . . 3 (𝐵 = ∅ → (𝑓:𝐴𝐵𝑓:𝐴⟶∅))
32mobidv 2549 . 2 (𝐵 = ∅ → (∃*𝑓 𝑓:𝐴𝐵 ↔ ∃*𝑓 𝑓:𝐴⟶∅))
41, 3mpbiri 257 1 (𝐵 = ∅ → ∃*𝑓 𝑓:𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  ∃*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:  mofsn2  46172  mofsssn  46173  mofmo  46174
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