Theorem uhgrsubgrself 29894

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 3935	. . 3 ⊢ (Vtx‘𝐺) ⊆ (Vtx‘𝐺)
2		ssid 3935	. . 3 ⊢ (iEdg‘𝐺) ⊆ (iEdg‘𝐺)
3	1, 2	pm3.2i 476	. 2 ⊢ ((Vtx‘𝐺) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝐺) ⊆ (iEdg‘𝐺))
4		eqid 2763	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
5	4	uhgrfun 29677	. . 3 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
6		id 23	. . 3 ⊢ (𝐺 ∈ UHGraph → 𝐺 ∈ UHGraph)
7		eqid 2763	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
8	7, 7, 4, 4	uhgrissubgr 29889	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ Fun (iEdg‘𝐺) ∧ 𝐺 ∈ UHGraph) → (𝐺 SubGraph 𝐺 ↔ ((Vtx‘𝐺) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝐺) ⊆ (iEdg‘𝐺))))
9	5, 6, 8	mpd3an23 1493	. 2 ⊢ (𝐺 ∈ UHGraph → (𝐺 SubGraph 𝐺 ↔ ((Vtx‘𝐺) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝐺) ⊆ (iEdg‘𝐺))))
10	3, 9	mpbiri 261	1 ⊢ (𝐺 ∈ UHGraph → 𝐺 SubGraph 𝐺)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2147   ⊆ wss 3900   class class class wbr 5106  Fun wfun 6555  ‘cfv 6561  Vtxcvtx 29607  iEdgciedg 29608  UHGraphcuhgr 29667   SubGraph csubgr 29881

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-edg 29659  df-uhgr 29669  df-subgr 29882

This theorem is used by: (None)