Theorem f00 6791

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 6740	. . . . 5 ⊢ (𝐹:𝐴⟶∅ → Fun 𝐹)
2		frn 6744	. . . . . . 7 ⊢ (𝐹:𝐴⟶∅ → ran 𝐹 ⊆ ∅)
3		ss0 4408	. . . . . . 7 ⊢ (ran 𝐹 ⊆ ∅ → ran 𝐹 = ∅)
4	2, 3	syl 17	. . . . . 6 ⊢ (𝐹:𝐴⟶∅ → ran 𝐹 = ∅)
5		dm0rn0 5938	. . . . . 6 ⊢ (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
6	4, 5	sylibr 234	. . . . 5 ⊢ (𝐹:𝐴⟶∅ → dom 𝐹 = ∅)
7		df-fn 6566	. . . . 5 ⊢ (𝐹 Fn ∅ ↔ (Fun 𝐹 ∧ dom 𝐹 = ∅))
8	1, 6, 7	sylanbrc 583	. . . 4 ⊢ (𝐹:𝐴⟶∅ → 𝐹 Fn ∅)
9		fn0 6700	. . . 4 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅)
10	8, 9	sylib 218	. . 3 ⊢ (𝐹:𝐴⟶∅ → 𝐹 = ∅)
11		fdm 6746	. . . 4 ⊢ (𝐹:𝐴⟶∅ → dom 𝐹 = 𝐴)
12	11, 6	eqtr3d 2777	. . 3 ⊢ (𝐹:𝐴⟶∅ → 𝐴 = ∅)
13	10, 12	jca 511	. 2 ⊢ (𝐹:𝐴⟶∅ → (𝐹 = ∅ ∧ 𝐴 = ∅))
14		f0 6790	. . 3 ⊢ ∅:∅⟶∅
15		feq1 6717	. . . 4 ⊢ (𝐹 = ∅ → (𝐹:𝐴⟶∅ ↔ ∅:𝐴⟶∅))
16		feq2 6718	. . . 4 ⊢ (𝐴 = ∅ → (∅:𝐴⟶∅ ↔ ∅:∅⟶∅))
17	15, 16	sylan9bb 509	. . 3 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → (𝐹:𝐴⟶∅ ↔ ∅:∅⟶∅))
18	14, 17	mpbiri 258	. 2 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → 𝐹:𝐴⟶∅)
19	13, 18	impbii 209	1 ⊢ (𝐹:𝐴⟶∅ ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537   ⊆ wss 3963  ∅c0 4339  dom cdm 5689  ran crn 5690  Fun wfun 6557   Fn wfn 6558  ⟶wf 6559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-fun 6565  df-fn 6566  df-f 6567

This theorem is referenced by:  dom0  9141  cantnff  9712  0wrd0  14575  supcvg  15889  ram0  17056  itgsubstlem  26104  uhgr0vb  29104  lfuhgr1v0e  29286  wlkv0  29684  sate0fv0  35402  prv0  35415  ismgmOLD  37837  mof0  48668  mof0ALT  48670  mofeu  48678  fdomne0  48680  f002  48684  fullthinc  48846