Theorem fset0 8860

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 6768	. . 3 ⊢ (𝑓:∅⟶𝐵 ↔ 𝑓 = ∅)
2	1	abbii 2833	. 2 ⊢ {𝑓 ∣ 𝑓:∅⟶𝐵} = {𝑓 ∣ 𝑓 = ∅}
3		df-sn 4595	. 2 ⊢ {∅} = {𝑓 ∣ 𝑓 = ∅}
4	2, 3	eqtr4i 2792	1 ⊢ {𝑓 ∣ 𝑓:∅⟶𝐵} = {∅}

This proof depends on syntax axioms:   = wceq 1570  {cab 2744  ∅c0 4289  {csn 4594  ⟶wf 6539

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2570  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-fun 6545  df-fn 6546  df-f 6547

This theorem is used by:  fsetexb  8870