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Theorem f00 6305
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 6262 . . . . 5 (𝐹:𝐴⟶∅ → Fun 𝐹)
2 frn 6265 . . . . . . 7 (𝐹:𝐴⟶∅ → ran 𝐹 ⊆ ∅)
3 ss0 4179 . . . . . . 7 (ran 𝐹 ⊆ ∅ → ran 𝐹 = ∅)
42, 3syl 17 . . . . . 6 (𝐹:𝐴⟶∅ → ran 𝐹 = ∅)
5 dm0rn0 5550 . . . . . 6 (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
64, 5sylibr 225 . . . . 5 (𝐹:𝐴⟶∅ → dom 𝐹 = ∅)
7 df-fn 6107 . . . . 5 (𝐹 Fn ∅ ↔ (Fun 𝐹 ∧ dom 𝐹 = ∅))
81, 6, 7sylanbrc 574 . . . 4 (𝐹:𝐴⟶∅ → 𝐹 Fn ∅)
9 fn0 6225 . . . 4 (𝐹 Fn ∅ ↔ 𝐹 = ∅)
108, 9sylib 209 . . 3 (𝐹:𝐴⟶∅ → 𝐹 = ∅)
11 fdm 6267 . . . 4 (𝐹:𝐴⟶∅ → dom 𝐹 = 𝐴)
1211, 6eqtr3d 2849 . . 3 (𝐹:𝐴⟶∅ → 𝐴 = ∅)
1310, 12jca 503 . 2 (𝐹:𝐴⟶∅ → (𝐹 = ∅ ∧ 𝐴 = ∅))
14 f0 6304 . . 3 ∅:∅⟶∅
15 feq1 6240 . . . 4 (𝐹 = ∅ → (𝐹:𝐴⟶∅ ↔ ∅:𝐴⟶∅))
16 feq2 6241 . . . 4 (𝐴 = ∅ → (∅:𝐴⟶∅ ↔ ∅:∅⟶∅))
1715, 16sylan9bb 501 . . 3 ((𝐹 = ∅ ∧ 𝐴 = ∅) → (𝐹:𝐴⟶∅ ↔ ∅:∅⟶∅))
1814, 17mpbiri 249 . 2 ((𝐹 = ∅ ∧ 𝐴 = ∅) → 𝐹:𝐴⟶∅)
1913, 18impbii 200 1 (𝐹:𝐴⟶∅ ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1637  wss 3776  c0 4123  dom cdm 5318  ran crn 5319  Fun wfun 6098   Fn wfn 6099  wf 6100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-sep 4982  ax-nul 4990  ax-pr 5103
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3400  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-nul 4124  df-if 4287  df-sn 4378  df-pr 4380  df-op 4384  df-br 4852  df-opab 4914  df-id 5226  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-fun 6106  df-fn 6107  df-f 6108
This theorem is referenced by:  cantnff  8821  0wrd0  13545  supcvg  14813  ram0  15946  itgsubstlem  24031  uhgr0vb  26187  lfuhgr1v0e  26368  wlkv0  26781  ismgmOLD  33962
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