Theorem cfsetssfset 46501

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 4054	. . 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 481	. . 3 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏) → 𝑓:𝐴⟶𝐵)
4	2, 3	mpgbir 1793	. 2 ⊢ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐵}
5	1, 4	eqsstri 4012	1 ⊢ 𝐹 ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐵}

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533  {cab 2702  ∀wral 3051  ∃wrex 3060   ⊆ wss 3945  ⟶wf 6543  ‘cfv 6547

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-11 2146  ax-12 2166  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ss 3962

This theorem is referenced by: (None)