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Theorem uhgrsubgrself 29363

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 3945	. . 3 ⊢ (Vtx‘𝐺) ⊆ (Vtx‘𝐺)
2		ssid 3945	. . 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 29149	. . 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 29358	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ Fun (iEdg‘𝐺) ∧ 𝐺 ∈ UHGraph) → (𝐺 SubGraph 𝐺 ↔ ((Vtx‘𝐺) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝐺) ⊆ (iEdg‘𝐺))))
9	5, 6, 8	mpd3an23 1466	. 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 2114 ⊆ wss 3890 class class class wbr 5086 Fun wfun 6486 ‘cfv 6492 Vtxcvtx 29079 iEdgciedg 29080 UHGraphcuhgr 29139 SubGraph csubgr 29350

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pr 5370 ax-un 7682

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rab 3391 df-v 3432 df-sbc 3730 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-br 5087 df-opab 5149 df-mpt 5168 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 29131 df-uhgr 29141 df-subgr 29351

This theorem is referenced by: (None)

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