Theorem chneq1 18674

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

Assertion

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

Proof of Theorem chneq1

Dummy variables 𝑥 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq 5110	. . . 4 ⊢ ( < = 𝑅 → ((𝑐‘(𝑥 − 1)) < (𝑐‘𝑥) ↔ (𝑐‘(𝑥 − 1))𝑅(𝑐‘𝑥)))
2	1	ralbidv 3187	. . 3 ⊢ ( < = 𝑅 → (∀𝑥 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑥 − 1)) < (𝑐‘𝑥) ↔ ∀𝑥 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑥 − 1))𝑅(𝑐‘𝑥)))
3	2	rabbidv 3422	. 2 ⊢ ( < = 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 18668	. 2 ⊢ ( < Chain 𝐴) = {𝑐 ∈ Word 𝐴 ∣ ∀𝑥 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑥 − 1)) < (𝑐‘𝑥)}
5		df-chn 18668	. 2 ⊢ (𝑅 Chain 𝐴) = {𝑐 ∈ Word 𝐴 ∣ ∀𝑥 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑥 − 1))𝑅(𝑐‘𝑥)}
6	3, 4, 5	3eqtr4g 2822	1 ⊢ ( < = 𝑅 → ( < Chain 𝐴) = (𝑅 Chain 𝐴))

This proof depends on syntax axioms:   → wi 4   = wceq 1569  ∀wral 3078  {crab 3415   ∖ cdif 3901  {csn 4588   class class class wbr 5108  dom cdm 5660  ‘cfv 6536  (class class class)co 7412  0cc0 11106  1c1 11107   − cmin 11447  Word cword 14557   Chain cchn 18667

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rab 3416  df-br 5109  df-chn 18668

This theorem is used by:  chneq12  18676