uhgrsubgrself - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > uhgrsubgrself		Structured version Visualization version GIF version

Theorem uhgrsubgrself 29297

Description: A hypergraph is a subgraph of itself. (Contributed by AV, 17-Nov-2020.) (Proof shortened by AV, 21-Nov-2020.)

**Assertion**
Ref	Expression
uhgrsubgrself	⊢ (𝐺 ∈ UHGraph → 𝐺 SubGraph 𝐺)

**Proof of Theorem uhgrsubgrself**
Step	Hyp	Ref	Expression
1		ssid 4006	. . 3 ⊢ (Vtx‘𝐺) ⊆ (Vtx‘𝐺)
2		ssid 4006	. . 3 ⊢ (iEdg‘𝐺) ⊆ (iEdg‘𝐺)
3	1, 2	pm3.2i 470	. 2 ⊢ ((Vtx‘𝐺) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝐺) ⊆ (iEdg‘𝐺))
4		eqid 2737	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
5	4	uhgrfun 29083	. . 3 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
6		id 22	. . 3 ⊢ (𝐺 ∈ UHGraph → 𝐺 ∈ UHGraph)
7		eqid 2737	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
8	7, 7, 4, 4	uhgrissubgr 29292	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ Fun (iEdg‘𝐺) ∧ 𝐺 ∈ UHGraph) → (𝐺 SubGraph 𝐺 ↔ ((Vtx‘𝐺) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝐺) ⊆ (iEdg‘𝐺))))
9	5, 6, 8	mpd3an23 1465	. 2 ⊢ (𝐺 ∈ UHGraph → (𝐺 SubGraph 𝐺 ↔ ((Vtx‘𝐺) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝐺) ⊆ (iEdg‘𝐺))))
10	3, 9	mpbiri 258	1 ⊢ (𝐺 ∈ UHGraph → 𝐺 SubGraph 𝐺)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ⊆ wss 3951 class class class wbr 5143 Fun wfun 6555 ‘cfv 6561 Vtxcvtx 29013 iEdgciedg 29014 UHGraphcuhgr 29073 SubGraph csubgr 29284

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-edg 29065 df-uhgr 29075 df-subgr 29285

This theorem is referenced by: (None)

W3C validator