Theorem f0bi 6706

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 6651	. . 3 ⊢ (𝐹:∅⟶𝑋 → 𝐹 Fn ∅)
2		fn0 6612	. . 3 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅)
3	1, 2	sylib 218	. 2 ⊢ (𝐹:∅⟶𝑋 → 𝐹 = ∅)
4		f0 6704	. . 3 ⊢ ∅:∅⟶𝑋
5		feq1 6629	. . 3 ⊢ (𝐹 = ∅ → (𝐹:∅⟶𝑋 ↔ ∅:∅⟶𝑋))
6	4, 5	mpbiri 258	. 2 ⊢ (𝐹 = ∅ → 𝐹:∅⟶𝑋)
7	3, 6	impbii 209	1 ⊢ (𝐹:∅⟶𝑋 ↔ 𝐹 = ∅)

Syntax hints:   ↔ wb 206   = wceq 1541  ∅c0 4283   Fn wfn 6476  ⟶wf 6477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2535  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-fun 6483  df-fn 6484  df-f 6485

This theorem is referenced by:  f0dom0  6707  mapdm0  8766  fset0  8778  0map0sn0  8809  griedg0ssusgr  29244  rgrusgrprc  29569  sticksstones11  42195  2ffzoeq  47364  f102g  48889  homf0  49047  0funcg2  49122  0funcALT  49126