Metamath Proof Explorer


Theorem rtrclreclem1

Description: The reflexive, transitive closure is indeed a closure. (Contributed by Drahflow, 12-Nov-2015) (Revised by RP, 30-May-2020) (Revised by AV, 12-Jul-2024)

Ref Expression
Hypothesis rtrclreclem1.1 ( 𝜑𝑅𝑉 )
Assertion rtrclreclem1 ( 𝜑𝑅 ⊆ ( t*rec ‘ 𝑅 ) )

Proof

Step Hyp Ref Expression
1 rtrclreclem1.1 ( 𝜑𝑅𝑉 )
2 1nn0 1 ∈ ℕ0
3 ssidd ( 𝜑𝑅𝑅 )
4 1 relexp1d ( 𝜑 → ( 𝑅𝑟 1 ) = 𝑅 )
5 3 4 sseqtrrd ( 𝜑𝑅 ⊆ ( 𝑅𝑟 1 ) )
6 oveq2 ( 𝑛 = 1 → ( 𝑅𝑟 𝑛 ) = ( 𝑅𝑟 1 ) )
7 6 sseq2d ( 𝑛 = 1 → ( 𝑅 ⊆ ( 𝑅𝑟 𝑛 ) ↔ 𝑅 ⊆ ( 𝑅𝑟 1 ) ) )
8 7 rspcev ( ( 1 ∈ ℕ0𝑅 ⊆ ( 𝑅𝑟 1 ) ) → ∃ 𝑛 ∈ ℕ0 𝑅 ⊆ ( 𝑅𝑟 𝑛 ) )
9 2 5 8 sylancr ( 𝜑 → ∃ 𝑛 ∈ ℕ0 𝑅 ⊆ ( 𝑅𝑟 𝑛 ) )
10 ssiun ( ∃ 𝑛 ∈ ℕ0 𝑅 ⊆ ( 𝑅𝑟 𝑛 ) → 𝑅 𝑛 ∈ ℕ0 ( 𝑅𝑟 𝑛 ) )
11 9 10 syl ( 𝜑𝑅 𝑛 ∈ ℕ0 ( 𝑅𝑟 𝑛 ) )
12 eqidd ( 𝜑 → ( 𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 ( 𝑟𝑟 𝑛 ) ) = ( 𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 ( 𝑟𝑟 𝑛 ) ) )
13 oveq1 ( 𝑟 = 𝑅 → ( 𝑟𝑟 𝑛 ) = ( 𝑅𝑟 𝑛 ) )
14 13 iuneq2d ( 𝑟 = 𝑅 𝑛 ∈ ℕ0 ( 𝑟𝑟 𝑛 ) = 𝑛 ∈ ℕ0 ( 𝑅𝑟 𝑛 ) )
15 14 adantl ( ( 𝜑𝑟 = 𝑅 ) → 𝑛 ∈ ℕ0 ( 𝑟𝑟 𝑛 ) = 𝑛 ∈ ℕ0 ( 𝑅𝑟 𝑛 ) )
16 1 elexd ( 𝜑𝑅 ∈ V )
17 nn0ex 0 ∈ V
18 ovex ( 𝑅𝑟 𝑛 ) ∈ V
19 17 18 iunex 𝑛 ∈ ℕ0 ( 𝑅𝑟 𝑛 ) ∈ V
20 19 a1i ( 𝜑 𝑛 ∈ ℕ0 ( 𝑅𝑟 𝑛 ) ∈ V )
21 12 15 16 20 fvmptd ( 𝜑 → ( ( 𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 ( 𝑟𝑟 𝑛 ) ) ‘ 𝑅 ) = 𝑛 ∈ ℕ0 ( 𝑅𝑟 𝑛 ) )
22 11 21 sseqtrrd ( 𝜑𝑅 ⊆ ( ( 𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 ( 𝑟𝑟 𝑛 ) ) ‘ 𝑅 ) )
23 df-rtrclrec t*rec = ( 𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 ( 𝑟𝑟 𝑛 ) )
24 fveq1 ( t*rec = ( 𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 ( 𝑟𝑟 𝑛 ) ) → ( t*rec ‘ 𝑅 ) = ( ( 𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 ( 𝑟𝑟 𝑛 ) ) ‘ 𝑅 ) )
25 24 sseq2d ( t*rec = ( 𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 ( 𝑟𝑟 𝑛 ) ) → ( 𝑅 ⊆ ( t*rec ‘ 𝑅 ) ↔ 𝑅 ⊆ ( ( 𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 ( 𝑟𝑟 𝑛 ) ) ‘ 𝑅 ) ) )
26 25 imbi2d ( t*rec = ( 𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 ( 𝑟𝑟 𝑛 ) ) → ( ( 𝜑𝑅 ⊆ ( t*rec ‘ 𝑅 ) ) ↔ ( 𝜑𝑅 ⊆ ( ( 𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 ( 𝑟𝑟 𝑛 ) ) ‘ 𝑅 ) ) ) )
27 23 26 ax-mp ( ( 𝜑𝑅 ⊆ ( t*rec ‘ 𝑅 ) ) ↔ ( 𝜑𝑅 ⊆ ( ( 𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 ( 𝑟𝑟 𝑛 ) ) ‘ 𝑅 ) ) )
28 22 27 mpbir ( 𝜑𝑅 ⊆ ( t*rec ‘ 𝑅 ) )