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Mathbox for Scott Fenton |
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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 34505 | . 2 ⊢ Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))) | |
2 | resss 5963 | . 2 ⊢ (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))) ⊆ 𝐹 | |
3 | 1, 2 | eqsstri 3979 | 1 ⊢ Funpart𝐹 ⊆ 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3444 ∩ cin 3910 ⊆ wss 3911 × cxp 5632 dom cdm 5634 ↾ cres 5636 ∘ ccom 5638 Singletoncsingle 34469 Singletons csingles 34470 Imagecimage 34471 Funpartcfunpart 34480 |
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-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3446 df-in 3918 df-ss 3928 df-res 5646 df-funpart 34505 |
This theorem is referenced by: (None) |
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