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| Mirrors > Home > MPE Home > Th. List > Mathboxes > ischn | Structured version Visualization version GIF version | ||
| Description: Property of being a chain. (Contributed by Thierry Arnoux, 19-Jun-2025.) |
| Ref | Expression |
|---|---|
| ischn | ⊢ (𝐶 ∈ ( < Chain𝐴) ↔ (𝐶 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dmeq 5912 | . . . 4 ⊢ (𝑐 = 𝐶 → dom 𝑐 = dom 𝐶) | |
| 2 | 1 | difeq1d 4124 | . . 3 ⊢ (𝑐 = 𝐶 → (dom 𝑐 ∖ {0}) = (dom 𝐶 ∖ {0})) |
| 3 | fveq1 6903 | . . . 4 ⊢ (𝑐 = 𝐶 → (𝑐‘(𝑛 − 1)) = (𝐶‘(𝑛 − 1))) | |
| 4 | fveq1 6903 | . . . 4 ⊢ (𝑐 = 𝐶 → (𝑐‘𝑛) = (𝐶‘𝑛)) | |
| 5 | 3, 4 | breq12d 5154 | . . 3 ⊢ (𝑐 = 𝐶 → ((𝑐‘(𝑛 − 1)) < (𝑐‘𝑛) ↔ (𝐶‘(𝑛 − 1)) < (𝐶‘𝑛))) |
| 6 | 2, 5 | raleqbidv 3345 | . 2 ⊢ (𝑐 = 𝐶 → (∀𝑛 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑛 − 1)) < (𝑐‘𝑛) ↔ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛))) |
| 7 | df-chn 32982 | . 2 ⊢ ( < Chain𝐴) = {𝑐 ∈ Word 𝐴 ∣ ∀𝑛 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑛 − 1)) < (𝑐‘𝑛)} | |
| 8 | 6, 7 | elrab2 3694 | 1 ⊢ (𝐶 ∈ ( < Chain𝐴) ↔ (𝐶 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛))) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3060 ∖ cdif 3947 {csn 4624 class class class wbr 5141 dom cdm 5683 ‘cfv 6559 (class class class)co 7429 0cc0 11151 1c1 11152 − cmin 11488 Word cword 14548 Chaincchn 32981 |
| 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 2007 ax-8 2110 ax-9 2118 ax-ext 2707 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-br 5142 df-dm 5693 df-iota 6512 df-fv 6567 df-chn 32982 |
| This theorem is referenced by: chnwrd 32984 chnltm1 32985 pfxchn 32986 chnind 32987 chnso 32990 |
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