Theorem chneq1 18748

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 5104	. . . 4 ⊢ ( < = 𝑅 → ((𝑐‘(𝑥 − 1)) < (𝑐‘𝑥) ↔ (𝑐‘(𝑥 − 1))𝑅(𝑐‘𝑥)))
2	1	ralbidv 3185	. . 3 ⊢ ( < = 𝑅 → (∀𝑥 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑥 − 1)) < (𝑐‘𝑥) ↔ ∀𝑥 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑥 − 1))𝑅(𝑐‘𝑥)))
3	2	rabbidv 3419	. 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 18742	. 2 ⊢ ( < Chain 𝐴) = {𝑐 ∈ Word 𝐴 ∣ ∀𝑥 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑥 − 1)) < (𝑐‘𝑥)}
5		df-chn 18742	. 2 ⊢ (𝑅 Chain 𝐴) = {𝑐 ∈ Word 𝐴 ∣ ∀𝑥 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑥 − 1))𝑅(𝑐‘𝑥)}
6	3, 4, 5	3eqtr4g 2820	1 ⊢ ( < = 𝑅 → ( < Chain 𝐴) = (𝑅 Chain 𝐴))

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ∀wral 3076  {crab 3412   ∖ cdif 3895  {csn 4583   class class class wbr 5102  dom cdm 5647  ‘cfv 6527  (class class class)co 7408  0cc0 11172  1c1 11173   − cmin 11513  Word cword 14626   Chain cchn 18741

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-br 5103  df-chn 18742

This theorem is used by:  chneq12  18750