Theorem funpartss 36526

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 36454	. 2 ⊢ Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
2		resss 5994	. 2 ⊢ (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))) ⊆ 𝐹
3	1, 2	eqsstri 3977	1 ⊢ Funpart𝐹 ⊆ 𝐹

This proof depends on syntax axioms:  Vcvv 3450   ∩ cin 3898   ⊆ wss 3899   × cxp 5653  dom cdm 5655   ↾ cres 5657   ∘ ccom 5659  Singletoncsingle 36418   Singletons csingles 36419  Imagecimage 36420  Funpartcfunpart 36429

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-in 3906  df-ss 3916  df-res 5667  df-funpart 36454

This theorem is used by: (None)