Theorem funpartss 36172

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 36100	. 2 ⊢ Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
2		resss 5953	. 2 ⊢ (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))) ⊆ 𝐹
3	1, 2	eqsstri 3961	1 ⊢ Funpart𝐹 ⊆ 𝐹

Syntax hints:  Vcvv 3431   ∩ cin 3882   ⊆ wss 3883   × cxp 5616  dom cdm 5618   ↾ cres 5620   ∘ ccom 5622  Singletoncsingle 36064   Singletons csingles 36065  Imagecimage 36066  Funpartcfunpart 36075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433  df-in 3890  df-ss 3900  df-res 5630  df-funpart 36100

This theorem is referenced by: (None)