Theorem funpartss 34746

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 34676	. 2 ⊢ Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
2		resss 5998	. 2 ⊢ (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))) ⊆ 𝐹
3	1, 2	eqsstri 4012	1 ⊢ Funpart𝐹 ⊆ 𝐹

Syntax hints:  Vcvv 3473   ∩ cin 3943   ⊆ wss 3944   × cxp 5667  dom cdm 5669   ↾ cres 5671   ∘ ccom 5673  Singletoncsingle 34640   Singletons csingles 34641  Imagecimage 34642  Funpartcfunpart 34651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-in 3951  df-ss 3961  df-res 5681  df-funpart 34676

This theorem is referenced by: (None)