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Theorem cfsetssfset 45752
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 4055 . . 3 ({𝑓 ∣ (𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏)} ⊆ {𝑓𝑓:𝐴𝐵} ↔ ∀𝑓((𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏) → 𝑓:𝐴𝐵))
3 simpl 483 . . 3 ((𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏) → 𝑓:𝐴𝐵)
42, 3mpgbir 1801 . 2 {𝑓 ∣ (𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏)} ⊆ {𝑓𝑓:𝐴𝐵}
51, 4eqsstri 4015 1 𝐹 ⊆ {𝑓𝑓:𝐴𝐵}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  {cab 2709  wral 3061  wrex 3070  wss 3947  wf 6536  cfv 6540
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-v 3476  df-in 3954  df-ss 3964
This theorem is referenced by: (None)
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