Theorem cfsetssfset 47410

Description: The class of constant functions is a subclass of the class of functions. (Contributed by AV, 13-Sep-2024.)

Hypothesis

Ref	Expression
cfsetsnfsetfv.f	⊢ 𝐹 = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)}

Assertion

Ref	Expression
cfsetssfset	⊢ 𝐹 ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐵}

Proof of Theorem cfsetssfset

Step	Hyp	Ref	Expression
1		cfsetsnfsetfv.f	. 2 ⊢ 𝐹 = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)}
2		ss2ab 4015	. . 3 ⊢ ({𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ↔ ∀𝑓((𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏) → 𝑓:𝐴⟶𝐵))
3		simpl 482	. . 3 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏) → 𝑓:𝐴⟶𝐵)
4	2, 3	mpgbir 1801	. 2 ⊢ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐵}
5	1, 4	eqsstri 3982	1 ⊢ 𝐹 ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐵}

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  {cab 2715  ∀wral 3052  ∃wrex 3062   ⊆ wss 3903  ⟶wf 6496  ‘cfv 6500

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-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ss 3920

This theorem is referenced by: (None)