Theorem ischn

Description: Property of being a chain. (Contributed by Thierry Arnoux, 19-Jun-2025)

	Ref	Expression
Assertion	ischn	⊢ ( 𝐶 ∈ ( < Chain 𝐴 ) ↔ ( 𝐶 ∈ Word 𝐴 ∧ ∀ 𝑛 ∈ ( dom 𝐶 ∖ { 0 } ) ( 𝐶 ‘ ( 𝑛 − 1 ) ) < ( 𝐶 ‘ 𝑛 ) ) )

Proof

Step	Hyp	Ref	Expression
1		dmeq	⊢ ( 𝑐 = 𝐶 → dom 𝑐 = dom 𝐶 )
2	1	difeq1d	⊢ ( 𝑐 = 𝐶 → ( dom 𝑐 ∖ { 0 } ) = ( dom 𝐶 ∖ { 0 } ) )
3		fveq1	⊢ ( 𝑐 = 𝐶 → ( 𝑐 ‘ ( 𝑛 − 1 ) ) = ( 𝐶 ‘ ( 𝑛 − 1 ) ) )
4		fveq1	⊢ ( 𝑐 = 𝐶 → ( 𝑐 ‘ 𝑛 ) = ( 𝐶 ‘ 𝑛 ) )
5	3 4	breq12d	⊢ ( 𝑐 = 𝐶 → ( ( 𝑐 ‘ ( 𝑛 − 1 ) ) < ( 𝑐 ‘ 𝑛 ) ↔ ( 𝐶 ‘ ( 𝑛 − 1 ) ) < ( 𝐶 ‘ 𝑛 ) ) )
6	2 5	raleqbidv	⊢ ( 𝑐 = 𝐶 → ( ∀ 𝑛 ∈ ( dom 𝑐 ∖ { 0 } ) ( 𝑐 ‘ ( 𝑛 − 1 ) ) < ( 𝑐 ‘ 𝑛 ) ↔ ∀ 𝑛 ∈ ( dom 𝐶 ∖ { 0 } ) ( 𝐶 ‘ ( 𝑛 − 1 ) ) < ( 𝐶 ‘ 𝑛 ) ) )
7		df-chn	⊢ ( < Chain 𝐴 ) = { 𝑐 ∈ Word 𝐴 ∣ ∀ 𝑛 ∈ ( dom 𝑐 ∖ { 0 } ) ( 𝑐 ‘ ( 𝑛 − 1 ) ) < ( 𝑐 ‘ 𝑛 ) }
8	6 7	elrab2	⊢ ( 𝐶 ∈ ( < Chain 𝐴 ) ↔ ( 𝐶 ∈ Word 𝐴 ∧ ∀ 𝑛 ∈ ( dom 𝐶 ∖ { 0 } ) ( 𝐶 ‘ ( 𝑛 − 1 ) ) < ( 𝐶 ‘ 𝑛 ) ) )