Theorem trclfv

Description: The transitive closure of a relation. (Contributed by RP, 28-Apr-2020)

	Ref	Expression
Assertion	trclfv	⊢ ( 𝑅 ∈ 𝑉 → ( t+ ‘ 𝑅 ) = ∩ { 𝑥 ∣ ( 𝑅 ⊆ 𝑥 ∧ ( 𝑥 ∘ 𝑥 ) ⊆ 𝑥 ) } )

Proof

Step	Hyp	Ref	Expression
1		elex	⊢ ( 𝑅 ∈ 𝑉 → 𝑅 ∈ V )
2		trclexlem	⊢ ( 𝑅 ∈ V → ( 𝑅 ∪ ( dom 𝑅 × ran 𝑅 ) ) ∈ V )
3		trclubg	⊢ ( 𝑅 ∈ V → ∩ { 𝑥 ∣ ( 𝑅 ⊆ 𝑥 ∧ ( 𝑥 ∘ 𝑥 ) ⊆ 𝑥 ) } ⊆ ( 𝑅 ∪ ( dom 𝑅 × ran 𝑅 ) ) )
4	2 3	ssexd	⊢ ( 𝑅 ∈ V → ∩ { 𝑥 ∣ ( 𝑅 ⊆ 𝑥 ∧ ( 𝑥 ∘ 𝑥 ) ⊆ 𝑥 ) } ∈ V )
5		trcleq1	⊢ ( 𝑟 = 𝑅 → ∩ { 𝑥 ∣ ( 𝑟 ⊆ 𝑥 ∧ ( 𝑥 ∘ 𝑥 ) ⊆ 𝑥 ) } = ∩ { 𝑥 ∣ ( 𝑅 ⊆ 𝑥 ∧ ( 𝑥 ∘ 𝑥 ) ⊆ 𝑥 ) } )
6		df-trcl	⊢ t+ = ( 𝑟 ∈ V ↦ ∩ { 𝑥 ∣ ( 𝑟 ⊆ 𝑥 ∧ ( 𝑥 ∘ 𝑥 ) ⊆ 𝑥 ) } )
7	5 6	fvmptg	⊢ ( ( 𝑅 ∈ V ∧ ∩ { 𝑥 ∣ ( 𝑅 ⊆ 𝑥 ∧ ( 𝑥 ∘ 𝑥 ) ⊆ 𝑥 ) } ∈ V ) → ( t+ ‘ 𝑅 ) = ∩ { 𝑥 ∣ ( 𝑅 ⊆ 𝑥 ∧ ( 𝑥 ∘ 𝑥 ) ⊆ 𝑥 ) } )
8	1 4 7	syl2anc2	⊢ ( 𝑅 ∈ 𝑉 → ( t+ ‘ 𝑅 ) = ∩ { 𝑥 ∣ ( 𝑅 ⊆ 𝑥 ∧ ( 𝑥 ∘ 𝑥 ) ⊆ 𝑥 ) } )