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Mirrors > Home > MPE Home > Th. List > Mathboxes > isacycgr1 | Structured version Visualization version GIF version |
Description: The property of being an acyclic graph. (Contributed by BTernaryTau, 11-Oct-2023.) |
Ref | Expression |
---|---|
isacycgr1 | ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ AcyclicGraph ↔ ∀𝑓∀𝑝(𝑓(Cycles‘𝐺)𝑝 → 𝑓 = ∅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6907 | . . . . 5 ⊢ (𝑔 = 𝐺 → (Cycles‘𝑔) = (Cycles‘𝐺)) | |
2 | 1 | breqd 5159 | . . . 4 ⊢ (𝑔 = 𝐺 → (𝑓(Cycles‘𝑔)𝑝 ↔ 𝑓(Cycles‘𝐺)𝑝)) |
3 | 2 | imbi1d 341 | . . 3 ⊢ (𝑔 = 𝐺 → ((𝑓(Cycles‘𝑔)𝑝 → 𝑓 = ∅) ↔ (𝑓(Cycles‘𝐺)𝑝 → 𝑓 = ∅))) |
4 | 3 | 2albidv 1921 | . 2 ⊢ (𝑔 = 𝐺 → (∀𝑓∀𝑝(𝑓(Cycles‘𝑔)𝑝 → 𝑓 = ∅) ↔ ∀𝑓∀𝑝(𝑓(Cycles‘𝐺)𝑝 → 𝑓 = ∅))) |
5 | dfacycgr1 35129 | . 2 ⊢ AcyclicGraph = {𝑔 ∣ ∀𝑓∀𝑝(𝑓(Cycles‘𝑔)𝑝 → 𝑓 = ∅)} | |
6 | 4, 5 | elab2g 3683 | 1 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ AcyclicGraph ↔ ∀𝑓∀𝑝(𝑓(Cycles‘𝐺)𝑝 → 𝑓 = ∅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∀wal 1535 = wceq 1537 ∈ wcel 2106 ∅c0 4339 class class class wbr 5148 ‘cfv 6563 Cyclesccycls 29818 AcyclicGraphcacycgr 35127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-acycgr 35128 |
This theorem is referenced by: acycgrcycl 35132 acycgr1v 35134 |
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