Theorem chneq1 18578

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 5088	. . . 4 ⊢ ( < = 𝑅 → ((𝑐‘(𝑥 − 1)) < (𝑐‘𝑥) ↔ (𝑐‘(𝑥 − 1))𝑅(𝑐‘𝑥)))
2	1	ralbidv 3161	. . 3 ⊢ ( < = 𝑅 → (∀𝑥 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑥 − 1)) < (𝑐‘𝑥) ↔ ∀𝑥 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑥 − 1))𝑅(𝑐‘𝑥)))
3	2	rabbidv 3397	. 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 18572	. 2 ⊢ ( < Chain 𝐴) = {𝑐 ∈ Word 𝐴 ∣ ∀𝑥 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑥 − 1)) < (𝑐‘𝑥)}
5		df-chn 18572	. 2 ⊢ (𝑅 Chain 𝐴) = {𝑐 ∈ Word 𝐴 ∣ ∀𝑥 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑥 − 1))𝑅(𝑐‘𝑥)}
6	3, 4, 5	3eqtr4g 2797	1 ⊢ ( < = 𝑅 → ( < Chain 𝐴) = (𝑅 Chain 𝐴))

Syntax hints:   → wi 4   = wceq 1542  ∀wral 3052  {crab 3390   ∖ cdif 3887  {csn 4568   class class class wbr 5086  dom cdm 5631  ‘cfv 6499  (class class class)co 7367  0cc0 11038  1c1 11039   − cmin 11377  Word cword 14475   Chain cchn 18571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rab 3391  df-br 5087  df-chn 18572

This theorem is referenced by:  chneq12  18580