Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > uspgrupgrushgr | Structured version Visualization version GIF version |
Description: A graph is a simple pseudograph iff it is a pseudograph and a simple hypergraph. (Contributed by AV, 30-Nov-2020.) |
Ref | Expression |
---|---|
uspgrupgrushgr | ⊢ (𝐺 ∈ USPGraph ↔ (𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uspgrupgr 40406 | . . 3 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph ) | |
2 | uspgrushgr 40405 | . . 3 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph ) | |
3 | 1, 2 | jca 553 | . 2 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph )) |
4 | eqid 2610 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
5 | eqid 2610 | . . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
6 | 4, 5 | ushgrf 25729 | . . . 4 ⊢ (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅})) |
7 | edgaval 25794 | . . . . 5 ⊢ (𝐺 ∈ UPGraph → (Edg‘𝐺) = ran (iEdg‘𝐺)) | |
8 | upgredgss 25806 | . . . . 5 ⊢ (𝐺 ∈ UPGraph → (Edg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}) | |
9 | 7, 8 | eqsstr3d 3603 | . . . 4 ⊢ (𝐺 ∈ UPGraph → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}) |
10 | f1ssr 6020 | . . . 4 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}) | |
11 | 6, 9, 10 | syl2anr 494 | . . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph ) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}) |
12 | 4, 5 | isuspgr 40382 | . . . 4 ⊢ (𝐺 ∈ UPGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})) |
13 | 12 | adantr 480 | . . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph ) → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})) |
14 | 11, 13 | mpbird 246 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph ) → 𝐺 ∈ USPGraph ) |
15 | 3, 14 | impbii 198 | 1 ⊢ (𝐺 ∈ USPGraph ↔ (𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph )) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∈ wcel 1977 {crab 2900 ∖ cdif 3537 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 {csn 4125 class class class wbr 4583 dom cdm 5038 ran crn 5039 –1-1→wf1 5801 ‘cfv 5804 ≤ cle 9954 2c2 10947 #chash 12979 Vtxcvtx 25673 iEdgciedg 25674 USHGraph cushgr 25723 UPGraph cupgr 25747 Edgcedga 25792 USPGraph cuspgr 40378 |
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-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-ral 2901 df-rex 2902 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-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-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fv 5812 df-ushgr 25725 df-upgr 25749 df-edga 25793 df-uspgr 40380 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |