Theorem chneq1 18787

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 18781	. 2 ⊢ ( < Chain 𝐴) = {𝑐 ∈ Word 𝐴 ∣ ∀𝑥 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑥 − 1)) < (𝑐‘𝑥)}
5		df-chn 18781	. 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 1569  ∀wral 3076  {crab 3412   ∖ cdif 4045  {csn 4583   class class class wbr 5102  dom cdm 5650  ‘cfv 6538  (class class class)co 7420  0cc0 11201  1c1 11202   − cmin 11542  Word cword 14659   Chain cchn 18780

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-br 5103  df-chn 18781

This theorem is used by:  chneq12  18789