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Theorem cfsetssfset 45766
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:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏)}
Assertion
Ref Expression
cfsetssfset 𝐹 ⊆ {𝑓𝑓:𝐴𝐵}

Proof of Theorem cfsetssfset
StepHypRef Expression
1 cfsetsnfsetfv.f . 2 𝐹 = {𝑓 ∣ (𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏)}
2 ss2ab 4057 . . 3 ({𝑓 ∣ (𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏)} ⊆ {𝑓𝑓:𝐴𝐵} ↔ ∀𝑓((𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏) → 𝑓:𝐴𝐵))
3 simpl 484 . . 3 ((𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏) → 𝑓:𝐴𝐵)
42, 3mpgbir 1802 . 2 {𝑓 ∣ (𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏)} ⊆ {𝑓𝑓:𝐴𝐵}
51, 4eqsstri 4017 1 𝐹 ⊆ {𝑓𝑓:𝐴𝐵}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  {cab 2710  wral 3062  wrex 3071  wss 3949  wf 6540  cfv 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-v 3477  df-in 3956  df-ss 3966
This theorem is referenced by: (None)
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