Theorem fset0 8877

Description: The set of functions from the empty set is the singleton containing the empty set. (Contributed by AV, 13-Sep-2024.)

Assertion

Ref	Expression
fset0	⊢ {𝑓 ∣ 𝑓:∅⟶𝐵} = {∅}

Proof of Theorem fset0

Step	Hyp	Ref	Expression
1		f0bi 6772	. . 3 ⊢ (𝑓:∅⟶𝐵 ↔ 𝑓 = ∅)
2	1	abbii 2801	. 2 ⊢ {𝑓 ∣ 𝑓:∅⟶𝐵} = {𝑓 ∣ 𝑓 = ∅}
3		df-sn 4609	. 2 ⊢ {∅} = {𝑓 ∣ 𝑓 = ∅}
4	2, 3	eqtr4i 2760	1 ⊢ {𝑓 ∣ 𝑓:∅⟶𝐵} = {∅}

Syntax hints:   = wceq 1539  {cab 2712  ∅c0 4315  {csn 4608  ⟶wf 6538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5278  ax-nul 5288  ax-pr 5414

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3421  df-v 3466  df-dif 3936  df-un 3938  df-ss 3950  df-nul 4316  df-if 4508  df-sn 4609  df-pr 4611  df-op 4615  df-br 5126  df-opab 5188  df-id 5560  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-fun 6544  df-fn 6545  df-f 6546

This theorem is referenced by:  fsetexb  8887