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Theorem uhgrsubgrself 29353

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 3956	. . 3 ⊢ (Vtx‘𝐺) ⊆ (Vtx‘𝐺)
2		ssid 3956	. . 3 ⊢ (iEdg‘𝐺) ⊆ (iEdg‘𝐺)
3	1, 2	pm3.2i 470	. 2 ⊢ ((Vtx‘𝐺) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝐺) ⊆ (iEdg‘𝐺))
4		eqid 2736	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
5	4	uhgrfun 29139	. . 3 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
6		id 22	. . 3 ⊢ (𝐺 ∈ UHGraph → 𝐺 ∈ UHGraph)
7		eqid 2736	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
8	7, 7, 4, 4	uhgrissubgr 29348	. . 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 2113 ⊆ wss 3901 class class class wbr 5098 Fun wfun 6486 ‘cfv 6492 Vtxcvtx 29069 iEdgciedg 29070 UHGraphcuhgr 29129 SubGraph csubgr 29340

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pr 5377 ax-un 7680

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rab 3400 df-v 3442 df-sbc 3741 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-br 5099 df-opab 5161 df-mpt 5180 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-fv 6500 df-edg 29121 df-uhgr 29131 df-subgr 29341

This theorem is referenced by: (None)

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