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Theorem f00 6248
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 6209 . . . . 5 (𝐹:𝐴⟶∅ → Fun 𝐹)
2 frn 6214 . . . . . . 7 (𝐹:𝐴⟶∅ → ran 𝐹 ⊆ ∅)
3 ss0 4117 . . . . . . 7 (ran 𝐹 ⊆ ∅ → ran 𝐹 = ∅)
42, 3syl 17 . . . . . 6 (𝐹:𝐴⟶∅ → ran 𝐹 = ∅)
5 dm0rn0 5497 . . . . . 6 (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
64, 5sylibr 224 . . . . 5 (𝐹:𝐴⟶∅ → dom 𝐹 = ∅)
7 df-fn 6052 . . . . 5 (𝐹 Fn ∅ ↔ (Fun 𝐹 ∧ dom 𝐹 = ∅))
81, 6, 7sylanbrc 701 . . . 4 (𝐹:𝐴⟶∅ → 𝐹 Fn ∅)
9 fn0 6172 . . . 4 (𝐹 Fn ∅ ↔ 𝐹 = ∅)
108, 9sylib 208 . . 3 (𝐹:𝐴⟶∅ → 𝐹 = ∅)
11 fdm 6212 . . . 4 (𝐹:𝐴⟶∅ → dom 𝐹 = 𝐴)
1211, 6eqtr3d 2796 . . 3 (𝐹:𝐴⟶∅ → 𝐴 = ∅)
1310, 12jca 555 . 2 (𝐹:𝐴⟶∅ → (𝐹 = ∅ ∧ 𝐴 = ∅))
14 f0 6247 . . 3 ∅:∅⟶∅
15 feq1 6187 . . . 4 (𝐹 = ∅ → (𝐹:𝐴⟶∅ ↔ ∅:𝐴⟶∅))
16 feq2 6188 . . . 4 (𝐴 = ∅ → (∅:𝐴⟶∅ ↔ ∅:∅⟶∅))
1715, 16sylan9bb 738 . . 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 1632  wss 3715  c0 4058  dom cdm 5266  ran crn 5267  Fun wfun 6043   Fn wfn 6044  wf 6045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-fun 6051  df-fn 6052  df-f 6053
This theorem is referenced by:  cantnff  8744  0wrd0  13517  supcvg  14787  ram0  15928  itgsubstlem  24010  uhgr0vb  26166  lfuhgr1v0e  26345  wlkv0  26757  ismgmOLD  33962
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