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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 4062 | . . 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 482 | . . 3 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏) → 𝑓:𝐴⟶𝐵) | |
| 4 | 2, 3 | mpgbir 1799 | . 2 ⊢ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐵} |
| 5 | 1, 4 | eqsstri 4030 | 1 ⊢ 𝐹 ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐵} |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 {cab 2714 ∀wral 3061 ∃wrex 3070 ⊆ wss 3951 ⟶wf 6557 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ss 3968 |
| This theorem is referenced by: (None) |
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