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Mirrors > Home > MPE Home > Th. List > Mathboxes > cfsetssfset | Structured version Visualization version GIF version |
Description: The class of constant functions is a subclass of the class of functions. (Contributed by AV, 13-Sep-2024.) |
Ref | Expression |
---|---|
cfsetsnfsetfv.f | ⊢ 𝐹 = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} |
Ref | Expression |
---|---|
cfsetssfset | ⊢ 𝐹 ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐵} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cfsetsnfsetfv.f | . 2 ⊢ 𝐹 = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} | |
2 | ss2ab 4057 | . . 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 484 | . . 3 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏) → 𝑓:𝐴⟶𝐵) | |
4 | 2, 3 | mpgbir 1802 | . 2 ⊢ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐵} |
5 | 1, 4 | eqsstri 4017 | 1 ⊢ 𝐹 ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐵} |
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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