Theorem f0bi 6717

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 6662	. . 3 ⊢ (𝐹:∅⟶𝑋 → 𝐹 Fn ∅)
2		fn0 6623	. . 3 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅)
3	1, 2	sylib 219	. 2 ⊢ (𝐹:∅⟶𝑋 → 𝐹 = ∅)
4		f0 6715	. . 3 ⊢ ∅:∅⟶𝑋
5		feq1 6640	. . 3 ⊢ (𝐹 = ∅ → (𝐹:∅⟶𝑋 ↔ ∅:∅⟶𝑋))
6	4, 5	mpbiri 259	. 2 ⊢ (𝐹 = ∅ → 𝐹:∅⟶𝑋)
7	3, 6	impbii 210	1 ⊢ (𝐹:∅⟶𝑋 ↔ 𝐹 = ∅)

Syntax hints:   ↔ wb 207   = wceq 1547  ∅c0 4268   Fn wfn 6487  ⟶wf 6488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-mo 2543  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-fun 6494  df-fn 6495  df-f 6496

This theorem is referenced by:  f0dom0  6718  mapdm0  8786  fset0  8798  0map0sn0  8830  griedg0ssusgr  29359  rgrusgrprc  29683  vieta  33771  sticksstones11  42648  2ffzoeq  47798  f102g  49349  homf0  49506  0funcg2  49581  0funcALT  49585