Theorem fset0 8788

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 6711	. . 3 ⊢ (𝑓:∅⟶𝐵 ↔ 𝑓 = ∅)
2	1	abbii 2796	. 2 ⊢ {𝑓 ∣ 𝑓:∅⟶𝐵} = {𝑓 ∣ 𝑓 = ∅}
3		df-sn 4580	. 2 ⊢ {∅} = {𝑓 ∣ 𝑓 = ∅}
4	2, 3	eqtr4i 2755	1 ⊢ {𝑓 ∣ 𝑓:∅⟶𝐵} = {∅}

Syntax hints:   = wceq 1540  {cab 2707  ∅c0 4286  {csn 4579  ⟶wf 6482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  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 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-fun 6488  df-fn 6489  df-f 6490

This theorem is referenced by:  fsetexb  8798