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Theorem uhgrsubgrself 29427

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 3958	. . 3 ⊢ (Vtx‘𝐺) ⊆ (Vtx‘𝐺)
2		ssid 3958	. . 3 ⊢ (iEdg‘𝐺) ⊆ (iEdg‘𝐺)
3	1, 2	pm3.2i 474	. 2 ⊢ ((Vtx‘𝐺) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝐺) ⊆ (iEdg‘𝐺))
4		eqid 2761	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
5	4	uhgrfun 29213	. . 3 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
6		id 22	. . 3 ⊢ (𝐺 ∈ UHGraph → 𝐺 ∈ UHGraph)
7		eqid 2761	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
8	7, 7, 4, 4	uhgrissubgr 29422	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ Fun (iEdg‘𝐺) ∧ 𝐺 ∈ UHGraph) → (𝐺 SubGraph 𝐺 ↔ ((Vtx‘𝐺) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝐺) ⊆ (iEdg‘𝐺))))
9	5, 6, 8	mpd3an23 1483	. 2 ⊢ (𝐺 ∈ UHGraph → (𝐺 SubGraph 𝐺 ↔ ((Vtx‘𝐺) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝐺) ⊆ (iEdg‘𝐺))))
10	3, 9	mpbiri 260	1 ⊢ (𝐺 ∈ UHGraph → 𝐺 SubGraph 𝐺)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 ∈ wcel 2141 ⊆ wss 3904 class class class wbr 5099 Fun wfun 6511 ‘cfv 6517 Vtxcvtx 29143 iEdgciedg 29144 UHGraphcuhgr 29203 SubGraph csubgr 29414

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5245 ax-nul 5255 ax-pr 5389 ax-un 7714

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3076 df-rex 3086 df-rab 3414 df-v 3455 df-sbc 3745 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-br 5100 df-opab 5162 df-mpt 5181 df-id 5540 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-fv 6525 df-edg 29195 df-uhgr 29205 df-subgr 29415

This theorem is referenced by: (None)

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