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 : 𝑃 –onto→ 𝑆 |
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 : 𝑃 ⟶ 𝑃 |
5 | ffn | ⊢ ( 𝑅 : 𝑃 ⟶ 𝑃 → 𝑅 Fn 𝑃 ) | |
6 | 4 5 | ax-mp | ⊢ 𝑅 Fn 𝑃 |
7 | dffn4 | ⊢ ( 𝑅 Fn 𝑃 ↔ 𝑅 : 𝑃 –onto→ ran 𝑅 ) | |
8 | 6 7 | mpbi | ⊢ 𝑅 : 𝑃 –onto→ ran 𝑅 |
9 | 1 2 | mstaval | ⊢ 𝑆 = ran 𝑅 |
10 | foeq3 | ⊢ ( 𝑆 = ran 𝑅 → ( 𝑅 : 𝑃 –onto→ 𝑆 ↔ 𝑅 : 𝑃 –onto→ ran 𝑅 ) ) | |
11 | 9 10 | ax-mp | ⊢ ( 𝑅 : 𝑃 –onto→ 𝑆 ↔ 𝑅 : 𝑃 –onto→ ran 𝑅 ) |
12 | 8 11 | mpbir | ⊢ 𝑅 : 𝑃 –onto→ 𝑆 |