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Theorem subgruhgredgd 26069
Description: An edge of a subgraph of a hypergraph is a nonempty subset of its vertices. (Contributed by AV, 17-Nov-2020.) (Revised by AV, 21-Nov-2020.)
Hypotheses
Ref Expression
subgruhgredgd.v 𝑉 = (Vtx‘𝑆)
subgruhgredgd.i 𝐼 = (iEdg‘𝑆)
subgruhgredgd.g (𝜑𝐺 ∈ UHGraph )
subgruhgredgd.s (𝜑𝑆 SubGraph 𝐺)
subgruhgredgd.x (𝜑𝑋 ∈ dom 𝐼)
Assertion
Ref Expression
subgruhgredgd (𝜑 → (𝐼𝑋) ∈ (𝒫 𝑉 ∖ {∅}))

Proof of Theorem subgruhgredgd
StepHypRef Expression
1 subgruhgredgd.s . . 3 (𝜑𝑆 SubGraph 𝐺)
2 subgruhgredgd.v . . . 4 𝑉 = (Vtx‘𝑆)
3 eqid 2621 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
4 subgruhgredgd.i . . . 4 𝐼 = (iEdg‘𝑆)
5 eqid 2621 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
6 eqid 2621 . . . 4 (Edg‘𝑆) = (Edg‘𝑆)
72, 3, 4, 5, 6subgrprop2 26059 . . 3 (𝑆 SubGraph 𝐺 → (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉))
81, 7syl 17 . 2 (𝜑 → (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉))
9 simpr3 1067 . . . 4 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (Edg‘𝑆) ⊆ 𝒫 𝑉)
10 subgruhgredgd.g . . . . . . . . 9 (𝜑𝐺 ∈ UHGraph )
11 subgruhgrfun 26067 . . . . . . . . 9 ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
1210, 1, 11syl2anc 692 . . . . . . . 8 (𝜑 → Fun (iEdg‘𝑆))
13 subgruhgredgd.x . . . . . . . . 9 (𝜑𝑋 ∈ dom 𝐼)
144dmeqi 5285 . . . . . . . . 9 dom 𝐼 = dom (iEdg‘𝑆)
1513, 14syl6eleq 2708 . . . . . . . 8 (𝜑𝑋 ∈ dom (iEdg‘𝑆))
1612, 15jca 554 . . . . . . 7 (𝜑 → (Fun (iEdg‘𝑆) ∧ 𝑋 ∈ dom (iEdg‘𝑆)))
1716adantr 481 . . . . . 6 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (Fun (iEdg‘𝑆) ∧ 𝑋 ∈ dom (iEdg‘𝑆)))
184fveq1i 6149 . . . . . . 7 (𝐼𝑋) = ((iEdg‘𝑆)‘𝑋)
19 fvelrn 6308 . . . . . . 7 ((Fun (iEdg‘𝑆) ∧ 𝑋 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑋) ∈ ran (iEdg‘𝑆))
2018, 19syl5eqel 2702 . . . . . 6 ((Fun (iEdg‘𝑆) ∧ 𝑋 ∈ dom (iEdg‘𝑆)) → (𝐼𝑋) ∈ ran (iEdg‘𝑆))
2117, 20syl 17 . . . . 5 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼𝑋) ∈ ran (iEdg‘𝑆))
22 subgrv 26055 . . . . . . . . 9 (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))
2322simpld 475 . . . . . . . 8 (𝑆 SubGraph 𝐺𝑆 ∈ V)
241, 23syl 17 . . . . . . 7 (𝜑𝑆 ∈ V)
25 edgval 25841 . . . . . . 7 (𝑆 ∈ V → (Edg‘𝑆) = ran (iEdg‘𝑆))
2624, 25syl 17 . . . . . 6 (𝜑 → (Edg‘𝑆) = ran (iEdg‘𝑆))
2726adantr 481 . . . . 5 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (Edg‘𝑆) = ran (iEdg‘𝑆))
2821, 27eleqtrrd 2701 . . . 4 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼𝑋) ∈ (Edg‘𝑆))
299, 28sseldd 3584 . . 3 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼𝑋) ∈ 𝒫 𝑉)
305uhgrfun 25857 . . . . . . 7 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
3110, 30syl 17 . . . . . 6 (𝜑 → Fun (iEdg‘𝐺))
3231adantr 481 . . . . 5 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → Fun (iEdg‘𝐺))
33 simpr2 1066 . . . . 5 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → 𝐼 ⊆ (iEdg‘𝐺))
3413adantr 481 . . . . 5 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → 𝑋 ∈ dom 𝐼)
35 funssfv 6166 . . . . . 6 ((Fun (iEdg‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ 𝑋 ∈ dom 𝐼) → ((iEdg‘𝐺)‘𝑋) = (𝐼𝑋))
3635eqcomd 2627 . . . . 5 ((Fun (iEdg‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ 𝑋 ∈ dom 𝐼) → (𝐼𝑋) = ((iEdg‘𝐺)‘𝑋))
3732, 33, 34, 36syl3anc 1323 . . . 4 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼𝑋) = ((iEdg‘𝐺)‘𝑋))
3810adantr 481 . . . . 5 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → 𝐺 ∈ UHGraph )
39 funfn 5877 . . . . . . 7 (Fun (iEdg‘𝐺) ↔ (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
4031, 39sylib 208 . . . . . 6 (𝜑 → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
4140adantr 481 . . . . 5 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
42 subgreldmiedg 26068 . . . . . . 7 ((𝑆 SubGraph 𝐺𝑋 ∈ dom (iEdg‘𝑆)) → 𝑋 ∈ dom (iEdg‘𝐺))
431, 15, 42syl2anc 692 . . . . . 6 (𝜑𝑋 ∈ dom (iEdg‘𝐺))
4443adantr 481 . . . . 5 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → 𝑋 ∈ dom (iEdg‘𝐺))
455uhgrn0 25858 . . . . 5 ((𝐺 ∈ UHGraph ∧ (iEdg‘𝐺) Fn dom (iEdg‘𝐺) ∧ 𝑋 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑋) ≠ ∅)
4638, 41, 44, 45syl3anc 1323 . . . 4 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → ((iEdg‘𝐺)‘𝑋) ≠ ∅)
4737, 46eqnetrd 2857 . . 3 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼𝑋) ≠ ∅)
48 eldifsn 4287 . . 3 ((𝐼𝑋) ∈ (𝒫 𝑉 ∖ {∅}) ↔ ((𝐼𝑋) ∈ 𝒫 𝑉 ∧ (𝐼𝑋) ≠ ∅))
4929, 47, 48sylanbrc 697 . 2 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼𝑋) ∈ (𝒫 𝑉 ∖ {∅}))
508, 49mpdan 701 1 (𝜑 → (𝐼𝑋) ∈ (𝒫 𝑉 ∖ {∅}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  Vcvv 3186  cdif 3552  wss 3555  c0 3891  𝒫 cpw 4130  {csn 4148   class class class wbr 4613  dom cdm 5074  ran crn 5075  Fun wfun 5841   Fn wfn 5842  cfv 5847  Vtxcvtx 25774  iEdgciedg 25775  Edgcedg 25839   UHGraph cuhgr 25847   SubGraph csubgr 26052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-edg 25840  df-uhgr 25849  df-subgr 26053
This theorem is referenced by:  subumgredg2  26070  subuhgr  26071  subupgr  26072
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