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Mirrors > Home > MPE Home > Th. List > numclwwlkovq | Structured version Visualization version GIF version |
Description: Value of operation Q, mapping a vertex v and a nonnegative integer n to the not closed walks v(0) ... v(n) of length n from a fixed vertex v = v(0). "Not closed" means v(n) =/= v(0). (Contributed by Alexander van der Vekens, 27-Sep-2018.) |
Ref | Expression |
---|---|
numclwwlk.c | ⊢ 𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛)) |
numclwwlk.f | ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣}) |
numclwwlk.g | ⊢ 𝐺 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) |
numclwwlk.q | ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)}) |
Ref | Expression |
---|---|
numclwwlkovq | ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑋𝑄𝑁) = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6103 | . . . 4 ⊢ (𝑛 = 𝑁 → ((𝑉 WWalksN 𝐸)‘𝑛) = ((𝑉 WWalksN 𝐸)‘𝑁)) | |
2 | 1 | adantl 481 | . . 3 ⊢ ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) → ((𝑉 WWalksN 𝐸)‘𝑛) = ((𝑉 WWalksN 𝐸)‘𝑁)) |
3 | eqeq2 2621 | . . . . 5 ⊢ (𝑣 = 𝑋 → ((𝑤‘0) = 𝑣 ↔ (𝑤‘0) = 𝑋)) | |
4 | 3 | adantr 480 | . . . 4 ⊢ ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) → ((𝑤‘0) = 𝑣 ↔ (𝑤‘0) = 𝑋)) |
5 | simpl 472 | . . . . 5 ⊢ ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) → 𝑣 = 𝑋) | |
6 | 5 | neeq2d 2842 | . . . 4 ⊢ ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) → (( lastS ‘𝑤) ≠ 𝑣 ↔ ( lastS ‘𝑤) ≠ 𝑋)) |
7 | 4, 6 | anbi12d 743 | . . 3 ⊢ ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) → (((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣) ↔ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋))) |
8 | 2, 7 | rabeqbidv 3168 | . 2 ⊢ ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) → {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)} = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)}) |
9 | numclwwlk.q | . 2 ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)}) | |
10 | fvex 6113 | . . 3 ⊢ ((𝑉 WWalksN 𝐸)‘𝑁) ∈ V | |
11 | 10 | rabex 4740 | . 2 ⊢ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)} ∈ V |
12 | 8, 9, 11 | ovmpt2a 6689 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑋𝑄𝑁) = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 {crab 2900 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 0cc0 9815 − cmin 10145 2c2 10947 ℕ0cn0 11169 ℤ≥cuz 11563 lastS clsw 13147 WWalksN cwwlkn 26206 ClWWalksN cclwwlkn 26277 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 |
This theorem is referenced by: numclwwlkqhash 26627 numclwwlk2lem1 26629 numclwlk2lem2f 26630 numclwlk2lem2f1o 26632 |
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