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Theorem f00 6125
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 6086 . . . . 5 (𝐹:𝐴⟶∅ → Fun 𝐹)
2 frn 6091 . . . . . . 7 (𝐹:𝐴⟶∅ → ran 𝐹 ⊆ ∅)
3 ss0 4007 . . . . . . 7 (ran 𝐹 ⊆ ∅ → ran 𝐹 = ∅)
42, 3syl 17 . . . . . 6 (𝐹:𝐴⟶∅ → ran 𝐹 = ∅)
5 dm0rn0 5374 . . . . . 6 (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
64, 5sylibr 224 . . . . 5 (𝐹:𝐴⟶∅ → dom 𝐹 = ∅)
7 df-fn 5929 . . . . 5 (𝐹 Fn ∅ ↔ (Fun 𝐹 ∧ dom 𝐹 = ∅))
81, 6, 7sylanbrc 699 . . . 4 (𝐹:𝐴⟶∅ → 𝐹 Fn ∅)
9 fn0 6049 . . . 4 (𝐹 Fn ∅ ↔ 𝐹 = ∅)
108, 9sylib 208 . . 3 (𝐹:𝐴⟶∅ → 𝐹 = ∅)
11 fdm 6089 . . . 4 (𝐹:𝐴⟶∅ → dom 𝐹 = 𝐴)
1211, 6eqtr3d 2687 . . 3 (𝐹:𝐴⟶∅ → 𝐴 = ∅)
1310, 12jca 553 . 2 (𝐹:𝐴⟶∅ → (𝐹 = ∅ ∧ 𝐴 = ∅))
14 f0 6124 . . 3 ∅:∅⟶∅
15 feq1 6064 . . . 4 (𝐹 = ∅ → (𝐹:𝐴⟶∅ ↔ ∅:𝐴⟶∅))
16 feq2 6065 . . . 4 (𝐴 = ∅ → (∅:𝐴⟶∅ ↔ ∅:∅⟶∅))
1715, 16sylan9bb 736 . . 3 ((𝐹 = ∅ ∧ 𝐴 = ∅) → (𝐹:𝐴⟶∅ ↔ ∅:∅⟶∅))
1814, 17mpbiri 248 . 2 ((𝐹 = ∅ ∧ 𝐴 = ∅) → 𝐹:𝐴⟶∅)
1913, 18impbii 199 1 (𝐹:𝐴⟶∅ ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1523  wss 3607  c0 3948  dom cdm 5143  ran crn 5144  Fun wfun 5920   Fn wfn 5921  wf 5922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-fun 5928  df-fn 5929  df-f 5930
This theorem is referenced by:  cantnff  8609  0wrd0  13363  supcvg  14632  ram0  15773  itgsubstlem  23856  uhgr0vb  26012  lfuhgr1v0e  26191  wlkv0  26603  ismgmOLD  33779
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