Theorem f0bi 6713

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 6658	. . 3 ⊢ (𝐹:∅⟶𝑋 → 𝐹 Fn ∅)
2		fn0 6619	. . 3 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅)
3	1, 2	sylib 218	. 2 ⊢ (𝐹:∅⟶𝑋 → 𝐹 = ∅)
4		f0 6711	. . 3 ⊢ ∅:∅⟶𝑋
5		feq1 6636	. . 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 4282   Fn wfn 6483  ⟶wf 6484

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 2115  ax-9 2123  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 2537  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-fun 6490  df-fn 6491  df-f 6492

This theorem is referenced by:  f0dom0  6714  mapdm0  8774  fset0  8786  0map0sn0  8817  griedg0ssusgr  29247  rgrusgrprc  29572  sticksstones11  42272  2ffzoeq  47454  f102g  48979  homf0  49137  0funcg2  49212  0funcALT  49216