Theorem f00 6656

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

Step	Hyp	Ref	Expression
1		ffun 6603	. . . . 5 ⊢ (𝐹:𝐴⟶∅ → Fun 𝐹)
2		frn 6607	. . . . . . 7 ⊢ (𝐹:𝐴⟶∅ → ran 𝐹 ⊆ ∅)
3		ss0 4332	. . . . . . 7 ⊢ (ran 𝐹 ⊆ ∅ → ran 𝐹 = ∅)
4	2, 3	syl 17	. . . . . 6 ⊢ (𝐹:𝐴⟶∅ → ran 𝐹 = ∅)
5		dm0rn0 5834	. . . . . 6 ⊢ (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
6	4, 5	sylibr 233	. . . . 5 ⊢ (𝐹:𝐴⟶∅ → dom 𝐹 = ∅)
7		df-fn 6436	. . . . 5 ⊢ (𝐹 Fn ∅ ↔ (Fun 𝐹 ∧ dom 𝐹 = ∅))
8	1, 6, 7	sylanbrc 583	. . . 4 ⊢ (𝐹:𝐴⟶∅ → 𝐹 Fn ∅)
9		fn0 6564	. . . 4 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅)
10	8, 9	sylib 217	. . 3 ⊢ (𝐹:𝐴⟶∅ → 𝐹 = ∅)
11		fdm 6609	. . . 4 ⊢ (𝐹:𝐴⟶∅ → dom 𝐹 = 𝐴)
12	11, 6	eqtr3d 2780	. . 3 ⊢ (𝐹:𝐴⟶∅ → 𝐴 = ∅)
13	10, 12	jca 512	. 2 ⊢ (𝐹:𝐴⟶∅ → (𝐹 = ∅ ∧ 𝐴 = ∅))
14		f0 6655	. . 3 ⊢ ∅:∅⟶∅
15		feq1 6581	. . . 4 ⊢ (𝐹 = ∅ → (𝐹:𝐴⟶∅ ↔ ∅:𝐴⟶∅))
16		feq2 6582	. . . 4 ⊢ (𝐴 = ∅ → (∅:𝐴⟶∅ ↔ ∅:∅⟶∅))
17	15, 16	sylan9bb 510	. . 3 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → (𝐹:𝐴⟶∅ ↔ ∅:∅⟶∅))
18	14, 17	mpbiri 257	. 2 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → 𝐹:𝐴⟶∅)
19	13, 18	impbii 208	1 ⊢ (𝐹:𝐴⟶∅ ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539   ⊆ wss 3887  ∅c0 4256  dom cdm 5589  ran crn 5590  Fun wfun 6427   Fn wfn 6428  ⟶wf 6429

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 5223  ax-nul 5230  ax-pr 5352

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-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-fun 6435  df-fn 6436  df-f 6437

This theorem is referenced by:  dom0  8889  cantnff  9432  0wrd0  14243  supcvg  15568  ram0  16723  itgsubstlem  25212  uhgr0vb  27442  lfuhgr1v0e  27621  wlkv0  28018  sate0fv0  33379  prv0  33392  ismgmOLD  36008  mof0  46165  mof0ALT  46167  mofeu  46175  fdomne0  46177  f002  46181  fullthinc  46327