Theorem chneq1

Description: Equality theorem for chains. (Contributed by Ender Ting, 17-Jan-2026)

	Ref	Expression
Assertion	chneq1	⊢ ( < = 𝑅 → ( < Chain 𝐴 ) = ( 𝑅 Chain 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		breq	⊢ ( < = 𝑅 → ( ( 𝑐 ‘ ( 𝑥 − 1 ) ) < ( 𝑐 ‘ 𝑥 ) ↔ ( 𝑐 ‘ ( 𝑥 − 1 ) ) 𝑅 ( 𝑐 ‘ 𝑥 ) ) )
2	1	ralbidv	⊢ ( < = 𝑅 → ( ∀ 𝑥 ∈ ( dom 𝑐 ∖ { 0 } ) ( 𝑐 ‘ ( 𝑥 − 1 ) ) < ( 𝑐 ‘ 𝑥 ) ↔ ∀ 𝑥 ∈ ( dom 𝑐 ∖ { 0 } ) ( 𝑐 ‘ ( 𝑥 − 1 ) ) 𝑅 ( 𝑐 ‘ 𝑥 ) ) )
3	2	rabbidv	⊢ ( < = 𝑅 → { 𝑐 ∈ Word 𝐴 ∣ ∀ 𝑥 ∈ ( dom 𝑐 ∖ { 0 } ) ( 𝑐 ‘ ( 𝑥 − 1 ) ) < ( 𝑐 ‘ 𝑥 ) } = { 𝑐 ∈ Word 𝐴 ∣ ∀ 𝑥 ∈ ( dom 𝑐 ∖ { 0 } ) ( 𝑐 ‘ ( 𝑥 − 1 ) ) 𝑅 ( 𝑐 ‘ 𝑥 ) } )
4		df-chn	⊢ ( < Chain 𝐴 ) = { 𝑐 ∈ Word 𝐴 ∣ ∀ 𝑥 ∈ ( dom 𝑐 ∖ { 0 } ) ( 𝑐 ‘ ( 𝑥 − 1 ) ) < ( 𝑐 ‘ 𝑥 ) }
5		df-chn	⊢ ( 𝑅 Chain 𝐴 ) = { 𝑐 ∈ Word 𝐴 ∣ ∀ 𝑥 ∈ ( dom 𝑐 ∖ { 0 } ) ( 𝑐 ‘ ( 𝑥 − 1 ) ) 𝑅 ( 𝑐 ‘ 𝑥 ) }
6	3 4 5	3eqtr4g	⊢ ( < = 𝑅 → ( < Chain 𝐴 ) = ( 𝑅 Chain 𝐴 ) )