Theorem f00 6757

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 6705	. . . . 5 ⊢ (𝐹:𝐴⟶∅ → Fun 𝐹)
2		frn 6710	. . . . . . 7 ⊢ (𝐹:𝐴⟶∅ → ran 𝐹 ⊆ ∅)
3		ss0 4352	. . . . . . 7 ⊢ (ran 𝐹 ⊆ ∅ → ran 𝐹 = ∅)
4	2, 3	syl 18	. . . . . 6 ⊢ (𝐹:𝐴⟶∅ → ran 𝐹 = ∅)
5		dm0rn0 5908	. . . . . 6 ⊢ (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
6	4, 5	sylibr 237	. . . . 5 ⊢ (𝐹:𝐴⟶∅ → dom 𝐹 = ∅)
7		df-fn 6536	. . . . 5 ⊢ (𝐹 Fn ∅ ↔ (Fun 𝐹 ∧ dom 𝐹 = ∅))
8	1, 6, 7	sylanbrc 595	. . . 4 ⊢ (𝐹:𝐴⟶∅ → 𝐹 Fn ∅)
9		fn0 6663	. . . 4 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅)
10	8, 9	sylib 221	. . 3 ⊢ (𝐹:𝐴⟶∅ → 𝐹 = ∅)
11		fdm 6712	. . . 4 ⊢ (𝐹:𝐴⟶∅ → dom 𝐹 = 𝐴)
12	11, 6	eqtr3d 2797	. . 3 ⊢ (𝐹:𝐴⟶∅ → 𝐴 = ∅)
13	10, 12	jca 521	. 2 ⊢ (𝐹:𝐴⟶∅ → (𝐹 = ∅ ∧ 𝐴 = ∅))
14		f0 6756	. . 3 ⊢ ∅:∅⟶∅
15		feq1 6680	. . . 4 ⊢ (𝐹 = ∅ → (𝐹:𝐴⟶∅ ↔ ∅:𝐴⟶∅))
16		feq2 6681	. . . 4 ⊢ (𝐴 = ∅ → (∅:𝐴⟶∅ ↔ ∅:∅⟶∅))
17	15, 16	sylan9bb 519	. . 3 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → (𝐹:𝐴⟶∅ ↔ ∅:∅⟶∅))
18	14, 17	mpbiri 261	. 2 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → 𝐹:𝐴⟶∅)
19	13, 18	impbii 212	1 ⊢ (𝐹:𝐴⟶∅ ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ⊆ wss 3899  ∅c0 4279  dom cdm 5655  ran crn 5656  Fun wfun 6527   Fn wfn 6528  ⟶wf 6529

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-fun 6535  df-fn 6536  df-f 6537

This theorem is used by:  dom0  9103  cantnff  9653  0wrd0  14605  supcvg  15945  ram0  17114  itgsubstlem  26275  uhgr0vb  29529  lfuhgr1v0e  29714  wlkv0  30109  sate0fv0  35996  prv0  36009  ismgmOLD  38600  mof0  49766  mof0ALT  49768  mofeu  49776  fdomne0  49778  f002  49782  fullthinc  50376