Theorem ischn 18653

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

Assertion

Ref	Expression
ischn	⊢ (𝐶 ∈ ( < Chain 𝐴) ↔ (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 5884	. . . 4 ⊢ (𝑐 = 𝐶 → dom 𝑐 = dom 𝐶)
2	1	difeq1d 4082	. . 3 ⊢ (𝑐 = 𝐶 → (dom 𝑐 ∖ {0}) = (dom 𝐶 ∖ {0}))
3		fveq1 6870	. . . 4 ⊢ (𝑐 = 𝐶 → (𝑐‘(𝑛 − 1)) = (𝐶‘(𝑛 − 1)))
4		fveq1 6870	. . . 4 ⊢ (𝑐 = 𝐶 → (𝑐‘𝑛) = (𝐶‘𝑛))
5	3, 4	breq12d 5118	. . 3 ⊢ (𝑐 = 𝐶 → ((𝑐‘(𝑛 − 1)) < (𝑐‘𝑛) ↔ (𝐶‘(𝑛 − 1)) < (𝐶‘𝑛)))
6	2, 5	raleqbidv 3339	. 2 ⊢ (𝑐 = 𝐶 → (∀𝑛 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑛 − 1)) < (𝑐‘𝑛) ↔ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛)))
7		df-chn 18652	. 2 ⊢ ( < Chain 𝐴) = {𝑐 ∈ Word 𝐴 ∣ ∀𝑛 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑛 − 1)) < (𝑐‘𝑛)}
8	6, 7	elrab2 3657	1 ⊢ (𝐶 ∈ ( < Chain 𝐴) ↔ (𝐶 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛)))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∀wral 3079   ∖ cdif 3904  {csn 4585   class class class wbr 5105  dom cdm 5652  ‘cfv 6525  (class class class)co 7400  0cc0 11088  1c1 11089   − cmin 11429  Word cword 14540   Chain cchn 18651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-dm 5662  df-iota 6481  df-fv 6533  df-chn 18652

This theorem is referenced by:  chnwrd  18654  chnltm1  18655  pfxchn  18656  chnrss  18661  chndss  18662  nulchn  18665  s1chn  18666  chnind  18667  chnso  18670  chnccats1  18671  chnccat  18672  chnrev  18673  ex-chn1  18683  ex-chn2  18684  chnsubseq  47454