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Theorem f002 46137
Description: A function with an empty codomain must have empty domain. (Contributed by Zhi Wang, 1-Oct-2024.)
Hypothesis
Ref Expression
f002.1 (𝜑𝐹:𝐴𝐵)
Assertion
Ref Expression
f002 (𝜑 → (𝐵 = ∅ → 𝐴 = ∅))

Proof of Theorem f002
StepHypRef Expression
1 f002.1 . 2 (𝜑𝐹:𝐴𝐵)
2 feq3 6576 . . 3 (𝐵 = ∅ → (𝐹:𝐴𝐵𝐹:𝐴⟶∅))
3 f00 6649 . . . 4 (𝐹:𝐴⟶∅ ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
43simprbi 497 . . 3 (𝐹:𝐴⟶∅ → 𝐴 = ∅)
52, 4syl6bi 252 . 2 (𝐵 = ∅ → (𝐹:𝐴𝐵𝐴 = ∅))
61, 5syl5com 31 1 (𝜑 → (𝐵 = ∅ → 𝐴 = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  c0 4257  wf 6423
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 5222  ax-nul 5229  ax-pr 5351
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-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3432  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4258  df-if 4461  df-sn 4563  df-pr 4565  df-op 4569  df-br 5075  df-opab 5137  df-id 5485  df-xp 5591  df-rel 5592  df-cnv 5593  df-co 5594  df-dm 5595  df-rn 5596  df-fun 6429  df-fn 6430  df-f 6431
This theorem is referenced by:  fullthinc2  46284  thincciso  46286
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