Theorem funpartss 36232

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.)

Assertion

Ref	Expression
funpartss	⊢ Funpart𝐹 ⊆ 𝐹

Proof of Theorem funpartss

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

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)