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Theorem subuhgr 40510
 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.
StepHypRef Expression
1 eqid 2610 . . . 4 (Vtx‘𝑆) = (Vtx‘𝑆)
2 eqid 2610 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
3 eqid 2610 . . . 4 (iEdg‘𝑆) = (iEdg‘𝑆)
4 eqid 2610 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
5 eqid 2610 . . . 4 (Edg‘𝑆) = (Edg‘𝑆)
61, 2, 3, 4, 5subgrprop2 40498 . . 3 (𝑆 SubGraph 𝐺 → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
7 subgruhgrfun 40506 . . . . . . . . 9 ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
87ancoms 468 . . . . . . . 8 ((𝑆 SubGraph 𝐺𝐺 ∈ UHGraph ) → Fun (iEdg‘𝑆))
98adantl 481 . . . . . . 7 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UHGraph )) → Fun (iEdg‘𝑆))
10 funfn 5833 . . . . . . 7 (Fun (iEdg‘𝑆) ↔ (iEdg‘𝑆) Fn dom (iEdg‘𝑆))
119, 10sylib 207 . . . . . 6 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UHGraph )) → (iEdg‘𝑆) Fn dom (iEdg‘𝑆))
12 simplrr 797 . . . . . . . . 9 (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UHGraph )) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → 𝐺 ∈ UHGraph )
13 simplrl 796 . . . . . . . . 9 (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UHGraph )) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → 𝑆 SubGraph 𝐺)
14 simpr 476 . . . . . . . . 9 (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UHGraph )) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → 𝑥 ∈ dom (iEdg‘𝑆))
151, 3, 12, 13, 14subgruhgredgd 40508 . . . . . . . 8 (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UHGraph )) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑥) ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}))
1615ralrimiva 2949 . . . . . . 7 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UHGraph )) → ∀𝑥 ∈ dom (iEdg‘𝑆)((iEdg‘𝑆)‘𝑥) ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}))
17 fnfvrnss 6297 . . . . . . 7 (((iEdg‘𝑆) Fn dom (iEdg‘𝑆) ∧ ∀𝑥 ∈ dom (iEdg‘𝑆)((iEdg‘𝑆)‘𝑥) ∈ (𝒫 (Vtx‘𝑆) ∖ {∅})) → ran (iEdg‘𝑆) ⊆ (𝒫 (Vtx‘𝑆) ∖ {∅}))
1811, 16, 17syl2anc 691 . . . . . 6 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UHGraph )) → ran (iEdg‘𝑆) ⊆ (𝒫 (Vtx‘𝑆) ∖ {∅}))
19 df-f 5808 . . . . . 6 ((iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅}) ↔ ((iEdg‘𝑆) Fn dom (iEdg‘𝑆) ∧ ran (iEdg‘𝑆) ⊆ (𝒫 (Vtx‘𝑆) ∖ {∅})))
2011, 18, 19sylanbrc 695 . . . . 5 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UHGraph )) → (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅}))
21 subgrv 40494 . . . . . . . 8 (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))
221, 3isuhgr 25726 . . . . . . . . 9 (𝑆 ∈ V → (𝑆 ∈ UHGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})))
2322adantr 480 . . . . . . . 8 ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (𝑆 ∈ UHGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})))
2421, 23syl 17 . . . . . . 7 (𝑆 SubGraph 𝐺 → (𝑆 ∈ UHGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})))
2524adantr 480 . . . . . 6 ((𝑆 SubGraph 𝐺𝐺 ∈ UHGraph ) → (𝑆 ∈ UHGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})))
2625adantl 481 . . . . 5 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UHGraph )) → (𝑆 ∈ UHGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶(𝒫 (Vtx‘𝑆) ∖ {∅})))
2720, 26mpbird 246 . . . 4 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UHGraph )) → 𝑆 ∈ UHGraph )
2827ex 449 . . 3 (((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) → ((𝑆 SubGraph 𝐺𝐺 ∈ UHGraph ) → 𝑆 ∈ UHGraph ))
296, 28syl 17 . 2 (𝑆 SubGraph 𝐺 → ((𝑆 SubGraph 𝐺𝐺 ∈ UHGraph ) → 𝑆 ∈ UHGraph ))
3029anabsi8 857 1 ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UHGraph )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125   class class class wbr 4583  dom cdm 5038  ran crn 5039  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  Vtxcvtx 25673  iEdgciedg 25674   UHGraph cuhgr 25722  Edgcedga 25792   SubGraph csubgr 40491 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-uhgr 25724  df-edga 25793  df-subgr 40492 This theorem is referenced by:  uhgrspan  40516
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