Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wwlknon | Structured version Visualization version GIF version |
Description: An element of the set of walks of a fixed length between two vertices as word. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 12-May-2021.) |
Ref | Expression |
---|---|
wwlknon.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
wwlknon | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ↔ (𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ (𝑊‘0) = 𝐴 ∧ (𝑊‘𝑁) = 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wwlknon.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | 1 | iswwlksnon 41051 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)}) |
3 | 2 | eleq2d 2673 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ↔ 𝑊 ∈ {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)})) |
4 | fveq1 6102 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑤‘0) = (𝑊‘0)) | |
5 | 4 | eqeq1d 2612 | . . . . 5 ⊢ (𝑤 = 𝑊 → ((𝑤‘0) = 𝐴 ↔ (𝑊‘0) = 𝐴)) |
6 | fveq1 6102 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑤‘𝑁) = (𝑊‘𝑁)) | |
7 | 6 | eqeq1d 2612 | . . . . 5 ⊢ (𝑤 = 𝑊 → ((𝑤‘𝑁) = 𝐵 ↔ (𝑊‘𝑁) = 𝐵)) |
8 | 5, 7 | anbi12d 743 | . . . 4 ⊢ (𝑤 = 𝑊 → (((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵) ↔ ((𝑊‘0) = 𝐴 ∧ (𝑊‘𝑁) = 𝐵))) |
9 | 8 | elrab 3331 | . . 3 ⊢ (𝑊 ∈ {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)} ↔ (𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘𝑁) = 𝐵))) |
10 | 3anass 1035 | . . 3 ⊢ ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ (𝑊‘0) = 𝐴 ∧ (𝑊‘𝑁) = 𝐵) ↔ (𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘𝑁) = 𝐵))) | |
11 | 9, 10 | bitr4i 266 | . 2 ⊢ (𝑊 ∈ {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)} ↔ (𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ (𝑊‘0) = 𝐴 ∧ (𝑊‘𝑁) = 𝐵)) |
12 | 3, 11 | syl6bb 275 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ↔ (𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ (𝑊‘0) = 𝐴 ∧ (𝑊‘𝑁) = 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 {crab 2900 ‘cfv 5804 (class class class)co 6549 0cc0 9815 Vtxcvtx 25673 WWalkSN cwwlksn 41029 WWalksNOn cwwlksnon 41030 |
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-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
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-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-wwlksn 41034 df-wwlksnon 41035 |
This theorem is referenced by: wwlksnwwlksnon 41121 wspthsnwspthsnon 41122 wspthsnonn0vne 41124 elwwlks2ons3 41159 s3wwlks2on 41160 wpthswwlks2on 41164 elwspths2spth 41171 |
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