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Theorem subgruhgredgd 27078
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 2801 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
4 subgruhgredgd.i . . . 4 𝐼 = (iEdg‘𝑆)
5 eqid 2801 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
6 eqid 2801 . . . 4 (Edg‘𝑆) = (Edg‘𝑆)
72, 3, 4, 5, 6subgrprop2 27068 . . 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 1193 . . . 4 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (Edg‘𝑆) ⊆ 𝒫 𝑉)
10 subgruhgredgd.g . . . . . . . . 9 (𝜑𝐺 ∈ UHGraph)
11 subgruhgrfun 27076 . . . . . . . . 9 ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
1210, 1, 11syl2anc 587 . . . . . . . 8 (𝜑 → Fun (iEdg‘𝑆))
13 subgruhgredgd.x . . . . . . . . 9 (𝜑𝑋 ∈ dom 𝐼)
144dmeqi 5741 . . . . . . . . 9 dom 𝐼 = dom (iEdg‘𝑆)
1513, 14eleqtrdi 2903 . . . . . . . 8 (𝜑𝑋 ∈ dom (iEdg‘𝑆))
1612, 15jca 515 . . . . . . 7 (𝜑 → (Fun (iEdg‘𝑆) ∧ 𝑋 ∈ dom (iEdg‘𝑆)))
1716adantr 484 . . . . . 6 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (Fun (iEdg‘𝑆) ∧ 𝑋 ∈ dom (iEdg‘𝑆)))
184fveq1i 6650 . . . . . . 7 (𝐼𝑋) = ((iEdg‘𝑆)‘𝑋)
19 fvelrn 6825 . . . . . . 7 ((Fun (iEdg‘𝑆) ∧ 𝑋 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑋) ∈ ran (iEdg‘𝑆))
2018, 19eqeltrid 2897 . . . . . 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 26846 . . . . 5 (Edg‘𝑆) = ran (iEdg‘𝑆)
2321, 22eleqtrrdi 2904 . . . 4 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼𝑋) ∈ (Edg‘𝑆))
249, 23sseldd 3919 . . 3 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼𝑋) ∈ 𝒫 𝑉)
255uhgrfun 26863 . . . . . . 7 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
2610, 25syl 17 . . . . . 6 (𝜑 → Fun (iEdg‘𝐺))
2726adantr 484 . . . . 5 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → Fun (iEdg‘𝐺))
28 simpr2 1192 . . . . 5 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → 𝐼 ⊆ (iEdg‘𝐺))
2913adantr 484 . . . . 5 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → 𝑋 ∈ dom 𝐼)
30 funssfv 6670 . . . . . 6 ((Fun (iEdg‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ 𝑋 ∈ dom 𝐼) → ((iEdg‘𝐺)‘𝑋) = (𝐼𝑋))
3130eqcomd 2807 . . . . 5 ((Fun (iEdg‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ 𝑋 ∈ dom 𝐼) → (𝐼𝑋) = ((iEdg‘𝐺)‘𝑋))
3227, 28, 29, 31syl3anc 1368 . . . 4 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼𝑋) = ((iEdg‘𝐺)‘𝑋))
3310adantr 484 . . . . 5 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → 𝐺 ∈ UHGraph)
3426funfnd 6359 . . . . . 6 (𝜑 → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
3534adantr 484 . . . . 5 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
36 subgreldmiedg 27077 . . . . . . 7 ((𝑆 SubGraph 𝐺𝑋 ∈ dom (iEdg‘𝑆)) → 𝑋 ∈ dom (iEdg‘𝐺))
371, 15, 36syl2anc 587 . . . . . 6 (𝜑𝑋 ∈ dom (iEdg‘𝐺))
3837adantr 484 . . . . 5 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → 𝑋 ∈ dom (iEdg‘𝐺))
395uhgrn0 26864 . . . . 5 ((𝐺 ∈ UHGraph ∧ (iEdg‘𝐺) Fn dom (iEdg‘𝐺) ∧ 𝑋 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑋) ≠ ∅)
4033, 35, 38, 39syl3anc 1368 . . . 4 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → ((iEdg‘𝐺)‘𝑋) ≠ ∅)
4132, 40eqnetrd 3057 . . 3 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼𝑋) ≠ ∅)
42 eldifsn 4683 . . 3 ((𝐼𝑋) ∈ (𝒫 𝑉 ∖ {∅}) ↔ ((𝐼𝑋) ∈ 𝒫 𝑉 ∧ (𝐼𝑋) ≠ ∅))
4324, 41, 42sylanbrc 586 . 2 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼𝑋) ∈ (𝒫 𝑉 ∖ {∅}))
448, 43mpdan 686 1 (𝜑 → (𝐼𝑋) ∈ (𝒫 𝑉 ∖ {∅}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2112  wne 2990  cdif 3881  wss 3884  c0 4246  𝒫 cpw 4500  {csn 4528   class class class wbr 5033  dom cdm 5523  ran crn 5524  Fun wfun 6322   Fn wfn 6323  cfv 6328  Vtxcvtx 26793  iEdgciedg 26794  Edgcedg 26844  UHGraphcuhgr 26853   SubGraph csubgr 27061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-edg 26845  df-uhgr 26855  df-subgr 27062
This theorem is referenced by:  subumgredg2  27079  subuhgr  27080  subupgr  27081
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