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| Mirrors > Home > MPE Home > Th. List > Mathboxes > funpartss | Structured version Visualization version GIF version | ||
| Description: The functional part of 𝐹 is a subset of 𝐹. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
| Ref | Expression |
|---|---|
| funpartss | ⊢ Funpart𝐹 ⊆ 𝐹 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-funpart 35875 | . 2 ⊢ Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))) | |
| 2 | resss 6019 | . 2 ⊢ (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))) ⊆ 𝐹 | |
| 3 | 1, 2 | eqsstri 4030 | 1 ⊢ Funpart𝐹 ⊆ 𝐹 |
| Colors of variables: wff setvar class |
| Syntax hints: Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 × cxp 5683 dom cdm 5685 ↾ cres 5687 ∘ ccom 5689 Singletoncsingle 35839 Singletons csingles 35840 Imagecimage 35841 Funpartcfunpart 35850 |
| 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-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-in 3958 df-ss 3968 df-res 5697 df-funpart 35875 |
| This theorem is referenced by: (None) |
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