Metamath Proof Explorer


Theorem frege109d

Description: If A contains all elements of U and all elements after those in U in the transitive closure of R , then the image under R of A is a subclass of A . Similar to Proposition 109 of Frege1879 p. 74. Compare with frege109 . (Contributed by RP, 15-Jul-2020)

Ref Expression
Hypotheses frege109d.r ( 𝜑𝑅 ∈ V )
frege109d.a ( 𝜑𝐴 = ( 𝑈 ∪ ( ( t+ ‘ 𝑅 ) “ 𝑈 ) ) )
Assertion frege109d ( 𝜑 → ( 𝑅𝐴 ) ⊆ 𝐴 )

Proof

Step Hyp Ref Expression
1 frege109d.r ( 𝜑𝑅 ∈ V )
2 frege109d.a ( 𝜑𝐴 = ( 𝑈 ∪ ( ( t+ ‘ 𝑅 ) “ 𝑈 ) ) )
3 trclfvlb ( 𝑅 ∈ V → 𝑅 ⊆ ( t+ ‘ 𝑅 ) )
4 imass1 ( 𝑅 ⊆ ( t+ ‘ 𝑅 ) → ( 𝑅𝑈 ) ⊆ ( ( t+ ‘ 𝑅 ) “ 𝑈 ) )
5 1 3 4 3syl ( 𝜑 → ( 𝑅𝑈 ) ⊆ ( ( t+ ‘ 𝑅 ) “ 𝑈 ) )
6 coss1 ( 𝑅 ⊆ ( t+ ‘ 𝑅 ) → ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) ⊆ ( ( t+ ‘ 𝑅 ) ∘ ( t+ ‘ 𝑅 ) ) )
7 1 3 6 3syl ( 𝜑 → ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) ⊆ ( ( t+ ‘ 𝑅 ) ∘ ( t+ ‘ 𝑅 ) ) )
8 trclfvcotrg ( ( t+ ‘ 𝑅 ) ∘ ( t+ ‘ 𝑅 ) ) ⊆ ( t+ ‘ 𝑅 )
9 7 8 sstrdi ( 𝜑 → ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) ⊆ ( t+ ‘ 𝑅 ) )
10 imass1 ( ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) ⊆ ( t+ ‘ 𝑅 ) → ( ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) “ 𝑈 ) ⊆ ( ( t+ ‘ 𝑅 ) “ 𝑈 ) )
11 9 10 syl ( 𝜑 → ( ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) “ 𝑈 ) ⊆ ( ( t+ ‘ 𝑅 ) “ 𝑈 ) )
12 5 11 unssd ( 𝜑 → ( ( 𝑅𝑈 ) ∪ ( ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) “ 𝑈 ) ) ⊆ ( ( t+ ‘ 𝑅 ) “ 𝑈 ) )
13 ssun2 ( ( t+ ‘ 𝑅 ) “ 𝑈 ) ⊆ ( 𝑈 ∪ ( ( t+ ‘ 𝑅 ) “ 𝑈 ) )
14 12 13 sstrdi ( 𝜑 → ( ( 𝑅𝑈 ) ∪ ( ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) “ 𝑈 ) ) ⊆ ( 𝑈 ∪ ( ( t+ ‘ 𝑅 ) “ 𝑈 ) ) )
15 2 imaeq2d ( 𝜑 → ( 𝑅𝐴 ) = ( 𝑅 “ ( 𝑈 ∪ ( ( t+ ‘ 𝑅 ) “ 𝑈 ) ) ) )
16 imaundi ( 𝑅 “ ( 𝑈 ∪ ( ( t+ ‘ 𝑅 ) “ 𝑈 ) ) ) = ( ( 𝑅𝑈 ) ∪ ( 𝑅 “ ( ( t+ ‘ 𝑅 ) “ 𝑈 ) ) )
17 imaco ( ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) “ 𝑈 ) = ( 𝑅 “ ( ( t+ ‘ 𝑅 ) “ 𝑈 ) )
18 17 eqcomi ( 𝑅 “ ( ( t+ ‘ 𝑅 ) “ 𝑈 ) ) = ( ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) “ 𝑈 )
19 18 uneq2i ( ( 𝑅𝑈 ) ∪ ( 𝑅 “ ( ( t+ ‘ 𝑅 ) “ 𝑈 ) ) ) = ( ( 𝑅𝑈 ) ∪ ( ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) “ 𝑈 ) )
20 16 19 eqtri ( 𝑅 “ ( 𝑈 ∪ ( ( t+ ‘ 𝑅 ) “ 𝑈 ) ) ) = ( ( 𝑅𝑈 ) ∪ ( ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) “ 𝑈 ) )
21 15 20 eqtrdi ( 𝜑 → ( 𝑅𝐴 ) = ( ( 𝑅𝑈 ) ∪ ( ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) “ 𝑈 ) ) )
22 14 21 2 3sstr4d ( 𝜑 → ( 𝑅𝐴 ) ⊆ 𝐴 )