Theorem chneq1 18535

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 5100	. . . 4 ⊢ ( < = 𝑅 → ((𝑐‘(𝑥 − 1)) < (𝑐‘𝑥) ↔ (𝑐‘(𝑥 − 1))𝑅(𝑐‘𝑥)))
2	1	ralbidv 3159	. . 3 ⊢ ( < = 𝑅 → (∀𝑥 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑥 − 1)) < (𝑐‘𝑥) ↔ ∀𝑥 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑥 − 1))𝑅(𝑐‘𝑥)))
3	2	rabbidv 3406	. 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 18529	. 2 ⊢ ( < Chain 𝐴) = {𝑐 ∈ Word 𝐴 ∣ ∀𝑥 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑥 − 1)) < (𝑐‘𝑥)}
5		df-chn 18529	. 2 ⊢ (𝑅 Chain 𝐴) = {𝑐 ∈ Word 𝐴 ∣ ∀𝑥 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑥 − 1))𝑅(𝑐‘𝑥)}
6	3, 4, 5	3eqtr4g 2796	1 ⊢ ( < = 𝑅 → ( < Chain 𝐴) = (𝑅 Chain 𝐴))

Syntax hints:   → wi 4   = wceq 1541  ∀wral 3051  {crab 3399   ∖ cdif 3898  {csn 4580   class class class wbr 5098  dom cdm 5624  ‘cfv 6492  (class class class)co 7358  0cc0 11026  1c1 11027   − cmin 11364  Word cword 14436   Chain cchn 18528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rab 3400  df-br 5099  df-chn 18529

This theorem is referenced by:  chneq12  18537