Theorem fn0 3611

Description: A function with empty domain is empty.

Assertion

Ref	Expression
fn0	⊢ (F Fn ∅ ↔ F = ∅)

Proof of Theorem fn0

Step	Hyp	Ref	Expression
1		fndm 3593	. . . . 5 ⊢ (F Fn ∅ → dom F = ∅)
2		noel 2287	. . . . . . . . . 10 ⊢ ¬ x ∈ ∅
3		eleq2 1538	. . . . . . . . . 10 ⊢ (dom F = ∅ → (x ∈ dom F ↔ x ∈ ∅))
4	2, 3	mtbiri 719	. . . . . . . . 9 ⊢ (dom F = ∅ → ¬ x ∈ dom F)
5		visset 1816	. . . . . . . . . . 11 ⊢ x ∈ V
6	5	eldm2 3314	. . . . . . . . . 10 ⊢ (x ∈ dom F ↔ ∃y⟨x, y⟩ ∈ F)
7	6	negbii 187	. . . . . . . . 9 ⊢ (¬ x ∈ dom F ↔ ¬ ∃y⟨x, y⟩ ∈ F)
8	4, 7	sylib 198	. . . . . . . 8 ⊢ (dom F = ∅ → ¬ ∃y⟨x, y⟩ ∈ F)
9		alnex 1035	. . . . . . . 8 ⊢ (∀y ¬ ⟨x, y⟩ ∈ F ↔ ¬ ∃y⟨x, y⟩ ∈ F)
10	8, 9	sylibr 200	. . . . . . 7 ⊢ (dom F = ∅ → ∀y ¬ ⟨x, y⟩ ∈ F)
11	10	19.21bi 1062	. . . . . 6 ⊢ (dom F = ∅ → ¬ ⟨x, y⟩ ∈ F)
12		noel 2287	. . . . . 6 ⊢ ¬ ⟨x, y⟩ ∈ ∅
13	11, 12	jctir 293	. . . . 5 ⊢ (dom F = ∅ → (¬ ⟨x, y⟩ ∈ F ⋀ ¬ ⟨x, y⟩ ∈ ∅))
14		pm5.21 679	. . . . 5 ⊢ ((¬ ⟨x, y⟩ ∈ F ⋀ ¬ ⟨x, y⟩ ∈ ∅) → (⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ ∅))
15	1, 13, 14	3syl 20	. . . 4 ⊢ (F Fn ∅ → (⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ ∅))
16	15	19.21aivv 1289	. . 3 ⊢ (F Fn ∅ → ∀x∀y(⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ ∅))
17		fnrel 3592	. . . . 5 ⊢ (F Fn ∅ → Rel F)
18		rel0 3278	. . . . 5 ⊢ Rel ∅
19	17, 18	jctir 293	. . . 4 ⊢ (F Fn ∅ → (Rel F ⋀ Rel ∅))
20		eqrel 3256	. . . 4 ⊢ ((Rel F ⋀ Rel ∅) → (F = ∅ ↔ ∀x∀y(⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ ∅)))
21	19, 20	syl 10	. . 3 ⊢ (F Fn ∅ → (F = ∅ ↔ ∀x∀y(⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ ∅)))
22	16, 21	mpbird 196	. 2 ⊢ (F Fn ∅ → F = ∅)
23		df-fn 3199	. . . 4 ⊢ (∅ Fn ∅ ↔ (Fun ∅ ⋀ dom ∅ = ∅))
24		fun0 3550	. . . 4 ⊢ Fun ∅
25		dm0 3329	. . . 4 ⊢ dom ∅ = ∅
26	23, 24, 25	mpbir2an 732	. . 3 ⊢ ∅ Fn ∅
27		fneq1 3588	. . 3 ⊢ (F = ∅ → (F Fn ∅ ↔ ∅ Fn ∅))
28	26, 27	mpbiri 194	. 2 ⊢ (F = ∅ → F Fn ∅)
29	22, 28	impbi 157	1 ⊢ (F Fn ∅ ↔ F = ∅)

Syntax hints:  ¬ wn 2   ↔ wb 146   ⋀ wa 223  ∀wal 956   = wceq 958   ∈ wcel 960  ∃wex 982  ∅c0 2283  ⟨cop 2415  dom cdm 3176  Rel wrel 3181  Fun wfun 3182   Fn wfn 3183

This theorem is referenced by:  f0 3662  f00 3663  f1o00 3720  fo00 3721  fconstfv 3855  map0e 4348  ixp0x 4365  hon0 9714

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-fun 3198  df-fn 3199

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