Theorem f00 2773

Description: A class is a function with empty codomain iff it and its domain are empty.

Assertion

Ref	Expression
f00	⊢ (F:A–→∅ ↔ (F = ∅ ∧ A = ∅))

Proof of Theorem f00

Step	Hyp	Ref	Expression
1		ffun 2754	. . . . . 6 ⊢ (F:A–→∅ → Fun F)
2		frn 2757	. . . . . . . 8 ⊢ (F:A–→∅ → ran F ⊆ ∅)
3		ss0 1727	. . . . . . . 8 ⊢ (ran F ⊆ ∅ → ran F = ∅)
4	2, 3	syl 12	. . . . . . 7 ⊢ (F:A–→∅ → ran F = ∅)
5		dm0rn0 2549	. . . . . . 7 ⊢ (dom F = ∅ ↔ ran F = ∅)
6	4, 5	sylibr 175	. . . . . 6 ⊢ (F:A–→∅ → dom F = ∅)
7	1, 6	jca 236	. . . . 5 ⊢ (F:A–→∅ → (Fun F ∧ dom F = ∅))
8		df-fn 2433	. . . . 5 ⊢ (F Fn ∅ ↔ (Fun F ∧ dom F = ∅))
9	7, 8	sylibr 175	. . . 4 ⊢ (F:A–→∅ → F Fn ∅)
10		fn0 2739	. . . 4 ⊢ (F Fn ∅ ↔ F = ∅)
11	9, 10	sylib 173	. . 3 ⊢ (F:A–→∅ → F = ∅)
12		fdm 2756	. . . 4 ⊢ (F:A–→∅ → dom F = A)
13	12, 6	eqtr3d 1130	. . 3 ⊢ (F:A–→∅ → A = ∅)
14	11, 13	jca 236	. 2 ⊢ (F:A–→∅ → (F = ∅ ∧ A = ∅))
15		f0 2772	. . 3 ⊢ ∅:∅–→∅
16		feq1 2748	. . . 4 ⊢ (F = ∅ → (F:A–→∅ ↔ ∅:A–→∅))
17		feq2 2749	. . . 4 ⊢ (A = ∅ → (∅:A–→∅ ↔ ∅:∅–→∅))
18	16, 17	sylan9bb 418	. . 3 ⊢ ((F = ∅ ∧ A = ∅) → (F:A–→∅ ↔ ∅:∅–→∅))
19	15, 18	mpbiri 169	. 2 ⊢ ((F = ∅ ∧ A = ∅) → F:A–→∅)
20	14, 19	impbi 139	1 ⊢ (F:A–→∅ ↔ (F = ∅ ∧ A = ∅))

Syntax hints:   ↔ wb 127   ∧ wa 196   = wceq 1091   ⊆ wss 1487  ∅c0 1707  dom cdm 2410  ran crn 2411  Fun wfun 2416   Fn wfn 2417  –→wf 2418

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-fun 2432  df-fn 2433  df-f 2434

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