Theorem cfsetssfset 48070

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 4009	. . 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 488	. . 3 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏) → 𝑓:𝐴⟶𝐵)
4	2, 3	mpgbir 1832	. 2 ⊢ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐵}
5	1, 4	eqsstri 3977	1 ⊢ 𝐹 ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐵}

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  {cab 2739  ∀wral 3077  ∃wrex 3087   ⊆ wss 3899  ⟶wf 6527  ‘cfv 6531

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ss 3916

This theorem is used by: (None)