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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 33232 | . 2 ⊢ Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))) | |
2 | resss 5871 | . 2 ⊢ (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))) ⊆ 𝐹 | |
3 | 1, 2 | eqsstri 3998 | 1 ⊢ Funpart𝐹 ⊆ 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3492 ∩ cin 3932 ⊆ wss 3933 × cxp 5546 dom cdm 5548 ↾ cres 5550 ∘ ccom 5552 Singletoncsingle 33196 Singletons csingles 33197 Imagecimage 33198 Funpartcfunpart 33207 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-in 3940 df-ss 3949 df-res 5560 df-funpart 33232 |
This theorem is referenced by: (None) |
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