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Theorem subgruhgredgd 26367
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 2752 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
4 subgruhgredgd.i . . . 4 𝐼 = (iEdg‘𝑆)
5 eqid 2752 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
6 eqid 2752 . . . 4 (Edg‘𝑆) = (Edg‘𝑆)
72, 3, 4, 5, 6subgrprop2 26357 . . 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 1235 . . . 4 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (Edg‘𝑆) ⊆ 𝒫 𝑉)
10 subgruhgredgd.g . . . . . . . . 9 (𝜑𝐺 ∈ UHGraph)
11 subgruhgrfun 26365 . . . . . . . . 9 ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
1210, 1, 11syl2anc 696 . . . . . . . 8 (𝜑 → Fun (iEdg‘𝑆))
13 subgruhgredgd.x . . . . . . . . 9 (𝜑𝑋 ∈ dom 𝐼)
144dmeqi 5472 . . . . . . . . 9 dom 𝐼 = dom (iEdg‘𝑆)
1513, 14syl6eleq 2841 . . . . . . . 8 (𝜑𝑋 ∈ dom (iEdg‘𝑆))
1612, 15jca 555 . . . . . . 7 (𝜑 → (Fun (iEdg‘𝑆) ∧ 𝑋 ∈ dom (iEdg‘𝑆)))
1716adantr 472 . . . . . 6 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (Fun (iEdg‘𝑆) ∧ 𝑋 ∈ dom (iEdg‘𝑆)))
184fveq1i 6345 . . . . . . 7 (𝐼𝑋) = ((iEdg‘𝑆)‘𝑋)
19 fvelrn 6507 . . . . . . 7 ((Fun (iEdg‘𝑆) ∧ 𝑋 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑋) ∈ ran (iEdg‘𝑆))
2018, 19syl5eqel 2835 . . . . . 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 edgval 26132 . . . . 5 (Edg‘𝑆) = ran (iEdg‘𝑆)
2321, 22syl6eleqr 2842 . . . 4 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼𝑋) ∈ (Edg‘𝑆))
249, 23sseldd 3737 . . 3 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼𝑋) ∈ 𝒫 𝑉)
255uhgrfun 26152 . . . . . . 7 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
2610, 25syl 17 . . . . . 6 (𝜑 → Fun (iEdg‘𝐺))
2726adantr 472 . . . . 5 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → Fun (iEdg‘𝐺))
28 simpr2 1233 . . . . 5 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → 𝐼 ⊆ (iEdg‘𝐺))
2913adantr 472 . . . . 5 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → 𝑋 ∈ dom 𝐼)
30 funssfv 6362 . . . . . 6 ((Fun (iEdg‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ 𝑋 ∈ dom 𝐼) → ((iEdg‘𝐺)‘𝑋) = (𝐼𝑋))
3130eqcomd 2758 . . . . 5 ((Fun (iEdg‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ 𝑋 ∈ dom 𝐼) → (𝐼𝑋) = ((iEdg‘𝐺)‘𝑋))
3227, 28, 29, 31syl3anc 1473 . . . 4 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼𝑋) = ((iEdg‘𝐺)‘𝑋))
3310adantr 472 . . . . 5 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → 𝐺 ∈ UHGraph)
34 funfn 6071 . . . . . . 7 (Fun (iEdg‘𝐺) ↔ (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
3526, 34sylib 208 . . . . . 6 (𝜑 → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
3635adantr 472 . . . . 5 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
37 subgreldmiedg 26366 . . . . . . 7 ((𝑆 SubGraph 𝐺𝑋 ∈ dom (iEdg‘𝑆)) → 𝑋 ∈ dom (iEdg‘𝐺))
381, 15, 37syl2anc 696 . . . . . 6 (𝜑𝑋 ∈ dom (iEdg‘𝐺))
3938adantr 472 . . . . 5 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → 𝑋 ∈ dom (iEdg‘𝐺))
405uhgrn0 26153 . . . . 5 ((𝐺 ∈ UHGraph ∧ (iEdg‘𝐺) Fn dom (iEdg‘𝐺) ∧ 𝑋 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑋) ≠ ∅)
4133, 36, 39, 40syl3anc 1473 . . . 4 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → ((iEdg‘𝐺)‘𝑋) ≠ ∅)
4232, 41eqnetrd 2991 . . 3 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼𝑋) ≠ ∅)
43 eldifsn 4454 . . 3 ((𝐼𝑋) ∈ (𝒫 𝑉 ∖ {∅}) ↔ ((𝐼𝑋) ∈ 𝒫 𝑉 ∧ (𝐼𝑋) ≠ ∅))
4424, 42, 43sylanbrc 701 . 2 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼𝑋) ∈ (𝒫 𝑉 ∖ {∅}))
458, 44mpdan 705 1 (𝜑 → (𝐼𝑋) ∈ (𝒫 𝑉 ∖ {∅}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1624  wcel 2131  wne 2924  cdif 3704  wss 3707  c0 4050  𝒫 cpw 4294  {csn 4313   class class class wbr 4796  dom cdm 5258  ran crn 5259  Fun wfun 6035   Fn wfn 6036  cfv 6041  Vtxcvtx 26065  iEdgciedg 26066  Edgcedg 26130  UHGraphcuhgr 26142   SubGraph csubgr 26350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-fv 6049  df-edg 26131  df-uhgr 26144  df-subgr 26351
This theorem is referenced by:  subumgredg2  26368  subuhgr  26369  subupgr  26370
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