Metamath Proof Explorer


Theorem frege96d

Description: If C follows A in the transitive closure of R and B follows C in R , then B follows A in the transitive closure of R . Similar to Proposition 96 of Frege1879 p. 71. Compare with frege96 . (Contributed by RP, 15-Jul-2020)

Ref Expression
Hypotheses frege96d.r ( 𝜑𝑅 ∈ V )
frege96d.a ( 𝜑𝐴 ∈ V )
frege96d.b ( 𝜑𝐵 ∈ V )
frege96d.c ( 𝜑𝐶 ∈ V )
frege96d.ac ( 𝜑𝐴 ( t+ ‘ 𝑅 ) 𝐶 )
frege96d.cb ( 𝜑𝐶 𝑅 𝐵 )
Assertion frege96d ( 𝜑𝐴 ( t+ ‘ 𝑅 ) 𝐵 )

Proof

Step Hyp Ref Expression
1 frege96d.r ( 𝜑𝑅 ∈ V )
2 frege96d.a ( 𝜑𝐴 ∈ V )
3 frege96d.b ( 𝜑𝐵 ∈ V )
4 frege96d.c ( 𝜑𝐶 ∈ V )
5 frege96d.ac ( 𝜑𝐴 ( t+ ‘ 𝑅 ) 𝐶 )
6 frege96d.cb ( 𝜑𝐶 𝑅 𝐵 )
7 brcogw ( ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V ) ∧ ( 𝐴 ( t+ ‘ 𝑅 ) 𝐶𝐶 𝑅 𝐵 ) ) → 𝐴 ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) 𝐵 )
8 2 3 4 5 6 7 syl32anc ( 𝜑𝐴 ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) 𝐵 )
9 trclfvlb ( 𝑅 ∈ V → 𝑅 ⊆ ( t+ ‘ 𝑅 ) )
10 coss1 ( 𝑅 ⊆ ( t+ ‘ 𝑅 ) → ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) ⊆ ( ( t+ ‘ 𝑅 ) ∘ ( t+ ‘ 𝑅 ) ) )
11 1 9 10 3syl ( 𝜑 → ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) ⊆ ( ( t+ ‘ 𝑅 ) ∘ ( t+ ‘ 𝑅 ) ) )
12 trclfvcotrg ( ( t+ ‘ 𝑅 ) ∘ ( t+ ‘ 𝑅 ) ) ⊆ ( t+ ‘ 𝑅 )
13 11 12 sstrdi ( 𝜑 → ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) ⊆ ( t+ ‘ 𝑅 ) )
14 13 ssbrd ( 𝜑 → ( 𝐴 ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) 𝐵𝐴 ( t+ ‘ 𝑅 ) 𝐵 ) )
15 8 14 mpd ( 𝜑𝐴 ( t+ ‘ 𝑅 ) 𝐵 )