Metamath Proof Explorer


Theorem cnvtrclfv

Description: The converse of the transitive closure is equal to the transitive closure of the converse relation. (Contributed by RP, 19-Jul-2020)

Ref Expression
Assertion cnvtrclfv ( 𝑅𝑉 ( t+ ‘ 𝑅 ) = ( t+ ‘ 𝑅 ) )

Proof

Step Hyp Ref Expression
1 elex ( 𝑅𝑉𝑅 ∈ V )
2 nnnn0 ( 𝑛 ∈ ℕ → 𝑛 ∈ ℕ0 )
3 relexpcnv ( ( 𝑛 ∈ ℕ0𝑅 ∈ V ) → ( 𝑅𝑟 𝑛 ) = ( 𝑅𝑟 𝑛 ) )
4 2 3 sylan ( ( 𝑛 ∈ ℕ ∧ 𝑅 ∈ V ) → ( 𝑅𝑟 𝑛 ) = ( 𝑅𝑟 𝑛 ) )
5 4 expcom ( 𝑅 ∈ V → ( 𝑛 ∈ ℕ → ( 𝑅𝑟 𝑛 ) = ( 𝑅𝑟 𝑛 ) ) )
6 5 ralrimiv ( 𝑅 ∈ V → ∀ 𝑛 ∈ ℕ ( 𝑅𝑟 𝑛 ) = ( 𝑅𝑟 𝑛 ) )
7 iuneq2 ( ∀ 𝑛 ∈ ℕ ( 𝑅𝑟 𝑛 ) = ( 𝑅𝑟 𝑛 ) → 𝑛 ∈ ℕ ( 𝑅𝑟 𝑛 ) = 𝑛 ∈ ℕ ( 𝑅𝑟 𝑛 ) )
8 6 7 syl ( 𝑅 ∈ V → 𝑛 ∈ ℕ ( 𝑅𝑟 𝑛 ) = 𝑛 ∈ ℕ ( 𝑅𝑟 𝑛 ) )
9 oveq1 ( 𝑟 = 𝑅 → ( 𝑟𝑟 𝑛 ) = ( 𝑅𝑟 𝑛 ) )
10 9 iuneq2d ( 𝑟 = 𝑅 𝑛 ∈ ℕ ( 𝑟𝑟 𝑛 ) = 𝑛 ∈ ℕ ( 𝑅𝑟 𝑛 ) )
11 dftrcl3 t+ = ( 𝑟 ∈ V ↦ 𝑛 ∈ ℕ ( 𝑟𝑟 𝑛 ) )
12 nnex ℕ ∈ V
13 ovex ( 𝑅𝑟 𝑛 ) ∈ V
14 12 13 iunex 𝑛 ∈ ℕ ( 𝑅𝑟 𝑛 ) ∈ V
15 10 11 14 fvmpt ( 𝑅 ∈ V → ( t+ ‘ 𝑅 ) = 𝑛 ∈ ℕ ( 𝑅𝑟 𝑛 ) )
16 15 cnveqd ( 𝑅 ∈ V → ( t+ ‘ 𝑅 ) = 𝑛 ∈ ℕ ( 𝑅𝑟 𝑛 ) )
17 cnviun 𝑛 ∈ ℕ ( 𝑅𝑟 𝑛 ) = 𝑛 ∈ ℕ ( 𝑅𝑟 𝑛 )
18 16 17 eqtrdi ( 𝑅 ∈ V → ( t+ ‘ 𝑅 ) = 𝑛 ∈ ℕ ( 𝑅𝑟 𝑛 ) )
19 cnvexg ( 𝑅 ∈ V → 𝑅 ∈ V )
20 oveq1 ( 𝑠 = 𝑅 → ( 𝑠𝑟 𝑛 ) = ( 𝑅𝑟 𝑛 ) )
21 20 iuneq2d ( 𝑠 = 𝑅 𝑛 ∈ ℕ ( 𝑠𝑟 𝑛 ) = 𝑛 ∈ ℕ ( 𝑅𝑟 𝑛 ) )
22 dftrcl3 t+ = ( 𝑠 ∈ V ↦ 𝑛 ∈ ℕ ( 𝑠𝑟 𝑛 ) )
23 ovex ( 𝑅𝑟 𝑛 ) ∈ V
24 12 23 iunex 𝑛 ∈ ℕ ( 𝑅𝑟 𝑛 ) ∈ V
25 21 22 24 fvmpt ( 𝑅 ∈ V → ( t+ ‘ 𝑅 ) = 𝑛 ∈ ℕ ( 𝑅𝑟 𝑛 ) )
26 19 25 syl ( 𝑅 ∈ V → ( t+ ‘ 𝑅 ) = 𝑛 ∈ ℕ ( 𝑅𝑟 𝑛 ) )
27 8 18 26 3eqtr4d ( 𝑅 ∈ V → ( t+ ‘ 𝑅 ) = ( t+ ‘ 𝑅 ) )
28 1 27 syl ( 𝑅𝑉 ( t+ ‘ 𝑅 ) = ( t+ ‘ 𝑅 ) )