Theorem f0bi 6741

Description: A function with empty domain is empty. (Contributed by Alexander van der Vekens, 30-Jun-2018.)

Assertion

Ref	Expression
f0bi	⊢ (𝐹:∅⟶𝑋 ↔ 𝐹 = ∅)

Proof of Theorem f0bi

Step	Hyp	Ref	Expression
1		ffn 6685	. . 3 ⊢ (𝐹:∅⟶𝑋 → 𝐹 Fn ∅)
2		fn0 6646	. . 3 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅)
3	1, 2	sylib 220	. 2 ⊢ (𝐹:∅⟶𝑋 → 𝐹 = ∅)
4		f0 6739	. . 3 ⊢ ∅:∅⟶𝑋
5		feq1 6663	. . 3 ⊢ (𝐹 = ∅ → (𝐹:∅⟶𝑋 ↔ ∅:∅⟶𝑋))
6	4, 5	mpbiri 260	. 2 ⊢ (𝐹 = ∅ → 𝐹:∅⟶𝑋)
7	3, 6	impbii 211	1 ⊢ (𝐹:∅⟶𝑋 ↔ 𝐹 = ∅)

Syntax hints:   ↔ wb 208   = wceq 1559  ∅c0 4283   Fn wfn 6510  ⟶wf 6511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-mo 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-fun 6517  df-fn 6518  df-f 6519

This theorem is referenced by:  f0dom0  6742  mapdm0  8816  fset0  8828  0map0sn0  8860  griedg0ssusgr  29422  rgrusgrprc  29746  vieta  33837  sticksstones11  42733  2ffzoeq  47882  f102g  49433  homf0  49590  0funcg2  49665  0funcALT  49669