Theorem fset0 8794

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 6717	. . 3 ⊢ (𝑓:∅⟶𝐵 ↔ 𝑓 = ∅)
2	1	abbii 2804	. 2 ⊢ {𝑓 ∣ 𝑓:∅⟶𝐵} = {𝑓 ∣ 𝑓 = ∅}
3		df-sn 4569	. 2 ⊢ {∅} = {𝑓 ∣ 𝑓 = ∅}
4	2, 3	eqtr4i 2763	1 ⊢ {𝑓 ∣ 𝑓:∅⟶𝐵} = {∅}

Syntax hints:   = wceq 1542  {cab 2715  ∅c0 4274  {csn 4568  ⟶wf 6488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-fun 6494  df-fn 6495  df-f 6496

This theorem is referenced by:  fsetexb  8804