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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 36160 | . 2 ⊢ Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))) | |
| 2 | resss 5976 | . 2 ⊢ (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))) ⊆ 𝐹 | |
| 3 | 1, 2 | eqsstri 3973 | 1 ⊢ Funpart𝐹 ⊆ 𝐹 |
| Colors of variables: wff setvar class |
| Syntax hints: Vcvv 3444 ∩ cin 3894 ⊆ wss 3895 × cxp 5634 dom cdm 5636 ↾ cres 5638 ∘ ccom 5640 Singletoncsingle 36124 Singletons csingles 36125 Imagecimage 36126 Funpartcfunpart 36135 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1805 ax-4 1819 ax-5 1920 ax-6 1977 ax-7 2018 ax-8 2134 ax-9 2142 ax-ext 2724 |
| This theorem depends on definitions: df-bi 209 df-an 399 df-tru 1553 df-ex 1790 df-sb 2081 df-clab 2731 df-cleq 2744 df-clel 2827 df-v 3446 df-in 3902 df-ss 3912 df-res 5648 df-funpart 36160 |
| This theorem is referenced by: (None) |
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