Theorem ischn 32961

Description: Property of being a chain. (Contributed by Thierry Arnoux, 19-Jun-2025.)

Assertion

Ref	Expression
ischn	⊢ (𝐶 ∈ ( < Chain𝐴) ↔ (𝐶 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛)))

Distinct variable groups:   < ,𝑛   𝐴,𝑛   𝐶,𝑛

Proof of Theorem ischn

Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmeq 5850	. . . 4 ⊢ (𝑐 = 𝐶 → dom 𝑐 = dom 𝐶)
2	1	difeq1d 4078	. . 3 ⊢ (𝑐 = 𝐶 → (dom 𝑐 ∖ {0}) = (dom 𝐶 ∖ {0}))
3		fveq1 6825	. . . 4 ⊢ (𝑐 = 𝐶 → (𝑐‘(𝑛 − 1)) = (𝐶‘(𝑛 − 1)))
4		fveq1 6825	. . . 4 ⊢ (𝑐 = 𝐶 → (𝑐‘𝑛) = (𝐶‘𝑛))
5	3, 4	breq12d 5108	. . 3 ⊢ (𝑐 = 𝐶 → ((𝑐‘(𝑛 − 1)) < (𝑐‘𝑛) ↔ (𝐶‘(𝑛 − 1)) < (𝐶‘𝑛)))
6	2, 5	raleqbidv 3310	. 2 ⊢ (𝑐 = 𝐶 → (∀𝑛 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑛 − 1)) < (𝑐‘𝑛) ↔ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛)))
7		df-chn 32960	. 2 ⊢ ( < Chain𝐴) = {𝑐 ∈ Word 𝐴 ∣ ∀𝑛 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑛 − 1)) < (𝑐‘𝑛)}
8	6, 7	elrab2 3653	1 ⊢ (𝐶 ∈ ( < Chain𝐴) ↔ (𝐶 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛)))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ∖ cdif 3902  {csn 4579   class class class wbr 5095  dom cdm 5623  ‘cfv 6486  (class class class)co 7353  0cc0 11028  1c1 11029   − cmin 11365  Word cword 14438  Chaincchn 32959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-dm 5633  df-iota 6442  df-fv 6494  df-chn 32960

This theorem is referenced by:  chnwrd  32962  chnltm1  32963  pfxchn  32964  s1chn  32965  chnind  32966  chnso  32969  chnccats1  32970