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Theorem f00 6000
 Description: A class is a function with empty codomain iff it and its domain are empty. (Contributed by NM, 10-Dec-2003.)
Assertion
Ref Expression
f00 (𝐹:𝐴⟶∅ ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))

Proof of Theorem f00
StepHypRef Expression
1 ffun 5961 . . . . 5 (𝐹:𝐴⟶∅ → Fun 𝐹)
2 frn 5966 . . . . . . 7 (𝐹:𝐴⟶∅ → ran 𝐹 ⊆ ∅)
3 ss0 3926 . . . . . . 7 (ran 𝐹 ⊆ ∅ → ran 𝐹 = ∅)
42, 3syl 17 . . . . . 6 (𝐹:𝐴⟶∅ → ran 𝐹 = ∅)
5 dm0rn0 5263 . . . . . 6 (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
64, 5sylibr 223 . . . . 5 (𝐹:𝐴⟶∅ → dom 𝐹 = ∅)
7 df-fn 5807 . . . . 5 (𝐹 Fn ∅ ↔ (Fun 𝐹 ∧ dom 𝐹 = ∅))
81, 6, 7sylanbrc 695 . . . 4 (𝐹:𝐴⟶∅ → 𝐹 Fn ∅)
9 fn0 5924 . . . 4 (𝐹 Fn ∅ ↔ 𝐹 = ∅)
108, 9sylib 207 . . 3 (𝐹:𝐴⟶∅ → 𝐹 = ∅)
11 fdm 5964 . . . 4 (𝐹:𝐴⟶∅ → dom 𝐹 = 𝐴)
1211, 6eqtr3d 2646 . . 3 (𝐹:𝐴⟶∅ → 𝐴 = ∅)
1310, 12jca 553 . 2 (𝐹:𝐴⟶∅ → (𝐹 = ∅ ∧ 𝐴 = ∅))
14 f0 5999 . . 3 ∅:∅⟶∅
15 feq1 5939 . . . 4 (𝐹 = ∅ → (𝐹:𝐴⟶∅ ↔ ∅:𝐴⟶∅))
16 feq2 5940 . . . 4 (𝐴 = ∅ → (∅:𝐴⟶∅ ↔ ∅:∅⟶∅))
1715, 16sylan9bb 732 . . 3 ((𝐹 = ∅ ∧ 𝐴 = ∅) → (𝐹:𝐴⟶∅ ↔ ∅:∅⟶∅))
1814, 17mpbiri 247 . 2 ((𝐹 = ∅ ∧ 𝐴 = ∅) → 𝐹:𝐴⟶∅)
1913, 18impbii 198 1 (𝐹:𝐴⟶∅ ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ⊆ wss 3540  ∅c0 3874  dom cdm 5038  ran crn 5039  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-fun 5806  df-fn 5807  df-f 5808 This theorem is referenced by:  cantnff  8454  0wrd0  13186  supcvg  14427  ram0  15564  itgsubstlem  23615  uhgr0vb  25738  uhgra0v  25839  usgra0v  25900  usgra1v  25919  wlkv0  26288  ismgmOLD  32819  lfuhgr1v0e  40480  1wlkv0  40859
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