Theorem fset0 8831

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 6743	. . 3 ⊢ (𝑓:∅⟶𝐵 ↔ 𝑓 = ∅)
2	1	abbii 2828	. 2 ⊢ {𝑓 ∣ 𝑓:∅⟶𝐵} = {𝑓 ∣ 𝑓 = ∅}
3		df-sn 4582	. 2 ⊢ {∅} = {𝑓 ∣ 𝑓 = ∅}
4	2, 3	eqtr4i 2787	1 ⊢ {𝑓 ∣ 𝑓:∅⟶𝐵} = {∅}

Syntax hints:   = wceq 1559  {cab 2739  ∅c0 4285  {csn 4581  ⟶wf 6513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-mo 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-fun 6519  df-fn 6520  df-f 6521

This theorem is referenced by:  fsetexb  8841