Theorem chneq2

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

	Ref	Expression
Assertion	chneq2	⊢ ( 𝐴 = 𝐵 → ( < Chain 𝐴 ) = ( < Chain 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		wrdeq	⊢ ( 𝐴 = 𝐵 → Word 𝐴 = Word 𝐵 )
2		rabeq	⊢ ( Word 𝐴 = Word 𝐵 → { 𝑐 ∈ Word 𝐴 ∣ ∀ 𝑥 ∈ ( dom 𝑐 ∖ { 0 } ) ( 𝑐 ‘ ( 𝑥 − 1 ) ) < ( 𝑐 ‘ 𝑥 ) } = { 𝑐 ∈ Word 𝐵 ∣ ∀ 𝑥 ∈ ( dom 𝑐 ∖ { 0 } ) ( 𝑐 ‘ ( 𝑥 − 1 ) ) < ( 𝑐 ‘ 𝑥 ) } )
3	1 2	syl	⊢ ( 𝐴 = 𝐵 → { 𝑐 ∈ Word 𝐴 ∣ ∀ 𝑥 ∈ ( dom 𝑐 ∖ { 0 } ) ( 𝑐 ‘ ( 𝑥 − 1 ) ) < ( 𝑐 ‘ 𝑥 ) } = { 𝑐 ∈ Word 𝐵 ∣ ∀ 𝑥 ∈ ( dom 𝑐 ∖ { 0 } ) ( 𝑐 ‘ ( 𝑥 − 1 ) ) < ( 𝑐 ‘ 𝑥 ) } )
4		df-chn	⊢ ( < Chain 𝐴 ) = { 𝑐 ∈ Word 𝐴 ∣ ∀ 𝑥 ∈ ( dom 𝑐 ∖ { 0 } ) ( 𝑐 ‘ ( 𝑥 − 1 ) ) < ( 𝑐 ‘ 𝑥 ) }
5		df-chn	⊢ ( < Chain 𝐵 ) = { 𝑐 ∈ Word 𝐵 ∣ ∀ 𝑥 ∈ ( dom 𝑐 ∖ { 0 } ) ( 𝑐 ‘ ( 𝑥 − 1 ) ) < ( 𝑐 ‘ 𝑥 ) }
6	3 4 5	3eqtr4g	⊢ ( 𝐴 = 𝐵 → ( < Chain 𝐴 ) = ( < Chain 𝐵 ) )