Metamath Proof Explorer


Theorem msrfo

Description: The reduct of a pre-statement is a statement. (Contributed by Mario Carneiro, 18-Jul-2016)

Ref Expression
Hypotheses mstaval.r 𝑅 = ( mStRed ‘ 𝑇 )
mstaval.s 𝑆 = ( mStat ‘ 𝑇 )
msrfo.p 𝑃 = ( mPreSt ‘ 𝑇 )
Assertion msrfo 𝑅 : 𝑃onto𝑆

Proof

Step Hyp Ref Expression
1 mstaval.r 𝑅 = ( mStRed ‘ 𝑇 )
2 mstaval.s 𝑆 = ( mStat ‘ 𝑇 )
3 msrfo.p 𝑃 = ( mPreSt ‘ 𝑇 )
4 3 1 msrf 𝑅 : 𝑃𝑃
5 ffn ( 𝑅 : 𝑃𝑃𝑅 Fn 𝑃 )
6 4 5 ax-mp 𝑅 Fn 𝑃
7 dffn4 ( 𝑅 Fn 𝑃𝑅 : 𝑃onto→ ran 𝑅 )
8 6 7 mpbi 𝑅 : 𝑃onto→ ran 𝑅
9 1 2 mstaval 𝑆 = ran 𝑅
10 foeq3 ( 𝑆 = ran 𝑅 → ( 𝑅 : 𝑃onto𝑆𝑅 : 𝑃onto→ ran 𝑅 ) )
11 9 10 ax-mp ( 𝑅 : 𝑃onto𝑆𝑅 : 𝑃onto→ ran 𝑅 )
12 8 11 mpbir 𝑅 : 𝑃onto𝑆