Theorem funpartss 36331

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 36259	. 2 ⊢ Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
2		resss 5998	. 2 ⊢ (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))) ⊆ 𝐹
3	1, 2	eqsstri 3991	1 ⊢ Funpart𝐹 ⊆ 𝐹

Syntax hints:  Vcvv 3463   ∩ cin 3912   ⊆ wss 3913   × cxp 5657  dom cdm 5659   ↾ cres 5661   ∘ ccom 5663  Singletoncsingle 36223   Singletons csingles 36224  Imagecimage 36225  Funpartcfunpart 36234

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-in 3920  df-ss 3930  df-res 5671  df-funpart 36259

This theorem is referenced by: (None)