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Mirrors > Home > MPE Home > Th. List > Mathboxes > wspthnp | Structured version Visualization version GIF version |
Description: Properties of a set being a simple path of a fixed length as word. (Contributed by AV, 18-May-2021.) |
Ref | Expression |
---|---|
wspthnp | ⊢ (𝑊 ∈ (𝑁 WSPathsN 𝐺) → ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ 𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ ∃𝑓 𝑓(SPathS‘𝐺)𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-wspthsn 41036 | . . 3 ⊢ WSPathsN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (𝑛 WWalkSN 𝑔) ∣ ∃𝑓 𝑓(SPathS‘𝑔)𝑤}) | |
2 | 1 | elmpt2cl 6774 | . 2 ⊢ (𝑊 ∈ (𝑁 WSPathsN 𝐺) → (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V)) |
3 | simpl 472 | . . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ 𝑊 ∈ (𝑁 WSPathsN 𝐺)) → (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V)) | |
4 | iswspthn 41047 | . . . . 5 ⊢ (𝑊 ∈ (𝑁 WSPathsN 𝐺) ↔ (𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ ∃𝑓 𝑓(SPathS‘𝐺)𝑊)) | |
5 | 4 | a1i 11 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝑊 ∈ (𝑁 WSPathsN 𝐺) ↔ (𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ ∃𝑓 𝑓(SPathS‘𝐺)𝑊))) |
6 | 5 | biimpa 500 | . . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ 𝑊 ∈ (𝑁 WSPathsN 𝐺)) → (𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ ∃𝑓 𝑓(SPathS‘𝐺)𝑊)) |
7 | 3anass 1035 | . . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ 𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ ∃𝑓 𝑓(SPathS‘𝐺)𝑊) ↔ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ ∃𝑓 𝑓(SPathS‘𝐺)𝑊))) | |
8 | 3, 6, 7 | sylanbrc 695 | . 2 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ 𝑊 ∈ (𝑁 WSPathsN 𝐺)) → ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ 𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ ∃𝑓 𝑓(SPathS‘𝐺)𝑊)) |
9 | 2, 8 | mpancom 700 | 1 ⊢ (𝑊 ∈ (𝑁 WSPathsN 𝐺) → ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ 𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ ∃𝑓 𝑓(SPathS‘𝐺)𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 ∃wex 1695 ∈ wcel 1977 {crab 2900 Vcvv 3173 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℕ0cn0 11169 SPathScspths 40920 WWalkSN cwwlksn 41029 WSPathsN cwwspthsn 41031 |
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-sep 4709 ax-nul 4717 ax-pow 4769 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 df-wwlksn 41034 df-wspthsn 41036 |
This theorem is referenced by: wspthsswwlkn 41125 |
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