Theorem subuhgr 29617

Description: A subgraph of a hypergraph is a hypergraph. (Contributed by AV, 16-Nov-2020.) (Proof shortened by AV, 21-Nov-2020.)

Assertion

Ref	Expression
subuhgr	⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UHGraph)

Proof of Theorem subuhgr

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2763	. . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
2		eqid 2763	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		eqid 2763	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
4		eqid 2763	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
5		eqid 2763	. . . 4 ⊢ (Edg‘𝑆) = (Edg‘𝑆)
6	1, 2, 3, 4, 5	subgrprop2 29605	. . 3 ⊢ (𝑆 SubGraph 𝐺 → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
7		subgruhgrfun 29613	. . . . . . . . 9 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
8	7	ancoms 463	. . . . . . . 8 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UHGraph) → Fun (iEdg‘𝑆))
9	8	adantl 486	. . . . . . 7 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UHGraph)) → Fun (iEdg‘𝑆))
10	9	funfnd 6569	. . . . . 6 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UHGraph)) → (iEdg‘𝑆) Fn dom (iEdg‘𝑆))
11		simplrr 789	. . . . . . . . 9 ⊢ (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UHGraph)) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → 𝐺 ∈ UHGraph)
12		simplrl 788	. . . . . . . . 9 ⊢ (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UHGraph)) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → 𝑆 SubGraph 𝐺)
13		simpr 489	. . . . . . . . 9 ⊢ (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UHGraph)) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → 𝑥 ∈ dom (iEdg‘𝑆))
14	1, 3, 11, 12, 13	subgruhgredgd 29615	. . . . . . . 8 ⊢ (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UHGraph)) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑥) ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}))
15	14	ralrimiva 3157	. . . . . . 7 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UHGraph)) → ∀𝑥 ∈ dom (iEdg‘𝑆)((iEdg‘𝑆)‘𝑥) ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}))
16		fnfvrnss 7118	. . . . . . 7 ⊢ (((iEdg‘𝑆) Fn dom (iEdg‘𝑆) ∧ ∀𝑥 ∈ dom (iEdg‘𝑆)((iEdg‘𝑆)‘𝑥) ∈ (𝒫 (Vtx‘𝑆) ∖ {∅})) → ran (iEdg‘𝑆) ⊆ (𝒫 (Vtx‘𝑆) ∖ {∅}))
17	10, 15, 16	syl2anc 595	. . . . . 6 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UHGraph)) → ran (iEdg‘𝑆) ⊆ (𝒫 (Vtx‘𝑆) ∖ {∅}))
18		df-f 6542	. . . . . 6 ⊢ ((iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅}) ↔ ((iEdg‘𝑆) Fn dom (iEdg‘𝑆) ∧ ran (iEdg‘𝑆) ⊆ (𝒫 (Vtx‘𝑆) ∖ {∅})))
19	10, 17, 18	sylanbrc 594	. . . . 5 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UHGraph)) → (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅}))
20		subgrv 29601	. . . . . . . 8 ⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))
21	1, 3	isuhgr 29391	. . . . . . . . 9 ⊢ (𝑆 ∈ V → (𝑆 ∈ UHGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})))
22	21	adantr 485	. . . . . . . 8 ⊢ ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (𝑆 ∈ UHGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})))
23	20, 22	syl 18	. . . . . . 7 ⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ UHGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})))
24	23	adantr 485	. . . . . 6 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UHGraph) → (𝑆 ∈ UHGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})))
25	24	adantl 486	. . . . 5 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UHGraph)) → (𝑆 ∈ UHGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})))
26	19, 25	mpbird 260	. . . 4 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UHGraph)) → 𝑆 ∈ UHGraph)
27	26	ex 417	. . 3 ⊢ (((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) → ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UHGraph) → 𝑆 ∈ UHGraph))
28	6, 27	syl 18	. 2 ⊢ (𝑆 SubGraph 𝐺 → ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UHGraph) → 𝑆 ∈ UHGraph))
29	28	anabsi8 684	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UHGraph)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   ∈ wcel 2143  ∀wral 3079  Vcvv 3455   ∖ cdif 3903   ⊆ wss 3906  ∅c0 4287  𝒫 cpw 4563  {csn 4590   class class class wbr 5110  dom cdm 5663  ran crn 5664  Fun wfun 6532   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  Vtxcvtx 29327  iEdgciedg 29328  Edgcedg 29378  UHGraphcuhgr 29387   SubGraph csubgr 29598

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-edg 29379  df-uhgr 29389  df-subgr 29599

This theorem is referenced by:  uhgrspan  29623