Theorem fsetabsnop 46495

Description: The class of all functions from a (proper) singleton into 𝐵 is the class of all the singletons of (proper) ordered pairs over the elements of 𝐵 as second component. (Contributed by AV, 13-Sep-2024.)

Assertion

Ref	Expression
fsetabsnop	⊢ (𝑆 ∈ 𝑉 → {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}})

Distinct variable groups:   𝐵,𝑏,𝑓   𝑆,𝑏,𝑓   𝑉,𝑏   𝑦,𝐵,𝑏   𝑦,𝑆

Allowed substitution hints:   𝑉(𝑦,𝑓)

Proof of Theorem fsetabsnop

Step	Hyp	Ref	Expression
1		fsetsniunop 46494	. 2 ⊢ (𝑆 ∈ 𝑉 → {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} = ∪ 𝑏 ∈ 𝐵 {{⟨𝑆, 𝑏⟩}})
2		iunsn 5064	. 2 ⊢ ∪ 𝑏 ∈ 𝐵 {{⟨𝑆, 𝑏⟩}} = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}}
3	1, 2	eqtrdi 2781	1 ⊢ (𝑆 ∈ 𝑉 → {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}})

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  {cab 2702  ∃wrex 3060  {csn 4624  ⟨cop 4630  ∪ ciun 4991  ⟶wf 6539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551

This theorem is referenced by:  fsetsnprcnex  46500