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Theorem subgruhgredgd 26221
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 2651 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
4 subgruhgredgd.i . . . 4 𝐼 = (iEdg‘𝑆)
5 eqid 2651 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
6 eqid 2651 . . . 4 (Edg‘𝑆) = (Edg‘𝑆)
72, 3, 4, 5, 6subgrprop2 26211 . . 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 1089 . . . 4 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (Edg‘𝑆) ⊆ 𝒫 𝑉)
10 subgruhgredgd.g . . . . . . . . 9 (𝜑𝐺 ∈ UHGraph)
11 subgruhgrfun 26219 . . . . . . . . 9 ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
1210, 1, 11syl2anc 694 . . . . . . . 8 (𝜑 → Fun (iEdg‘𝑆))
13 subgruhgredgd.x . . . . . . . . 9 (𝜑𝑋 ∈ dom 𝐼)
144dmeqi 5357 . . . . . . . . 9 dom 𝐼 = dom (iEdg‘𝑆)
1513, 14syl6eleq 2740 . . . . . . . 8 (𝜑𝑋 ∈ dom (iEdg‘𝑆))
1612, 15jca 553 . . . . . . 7 (𝜑 → (Fun (iEdg‘𝑆) ∧ 𝑋 ∈ dom (iEdg‘𝑆)))
1716adantr 480 . . . . . 6 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (Fun (iEdg‘𝑆) ∧ 𝑋 ∈ dom (iEdg‘𝑆)))
184fveq1i 6230 . . . . . . 7 (𝐼𝑋) = ((iEdg‘𝑆)‘𝑋)
19 fvelrn 6392 . . . . . . 7 ((Fun (iEdg‘𝑆) ∧ 𝑋 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑋) ∈ ran (iEdg‘𝑆))
2018, 19syl5eqel 2734 . . . . . 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 25986 . . . . 5 (Edg‘𝑆) = ran (iEdg‘𝑆)
2321, 22syl6eleqr 2741 . . . 4 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼𝑋) ∈ (Edg‘𝑆))
249, 23sseldd 3637 . . 3 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼𝑋) ∈ 𝒫 𝑉)
255uhgrfun 26006 . . . . . . 7 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
2610, 25syl 17 . . . . . 6 (𝜑 → Fun (iEdg‘𝐺))
2726adantr 480 . . . . 5 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → Fun (iEdg‘𝐺))
28 simpr2 1088 . . . . 5 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → 𝐼 ⊆ (iEdg‘𝐺))
2913adantr 480 . . . . 5 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → 𝑋 ∈ dom 𝐼)
30 funssfv 6247 . . . . . 6 ((Fun (iEdg‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ 𝑋 ∈ dom 𝐼) → ((iEdg‘𝐺)‘𝑋) = (𝐼𝑋))
3130eqcomd 2657 . . . . 5 ((Fun (iEdg‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ 𝑋 ∈ dom 𝐼) → (𝐼𝑋) = ((iEdg‘𝐺)‘𝑋))
3227, 28, 29, 31syl3anc 1366 . . . 4 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼𝑋) = ((iEdg‘𝐺)‘𝑋))
3310adantr 480 . . . . 5 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → 𝐺 ∈ UHGraph)
34 funfn 5956 . . . . . . 7 (Fun (iEdg‘𝐺) ↔ (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
3526, 34sylib 208 . . . . . 6 (𝜑 → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
3635adantr 480 . . . . 5 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
37 subgreldmiedg 26220 . . . . . . 7 ((𝑆 SubGraph 𝐺𝑋 ∈ dom (iEdg‘𝑆)) → 𝑋 ∈ dom (iEdg‘𝐺))
381, 15, 37syl2anc 694 . . . . . 6 (𝜑𝑋 ∈ dom (iEdg‘𝐺))
3938adantr 480 . . . . 5 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → 𝑋 ∈ dom (iEdg‘𝐺))
405uhgrn0 26007 . . . . 5 ((𝐺 ∈ UHGraph ∧ (iEdg‘𝐺) Fn dom (iEdg‘𝐺) ∧ 𝑋 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑋) ≠ ∅)
4133, 36, 39, 40syl3anc 1366 . . . 4 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → ((iEdg‘𝐺)‘𝑋) ≠ ∅)
4232, 41eqnetrd 2890 . . 3 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼𝑋) ≠ ∅)
43 eldifsn 4350 . . 3 ((𝐼𝑋) ∈ (𝒫 𝑉 ∖ {∅}) ↔ ((𝐼𝑋) ∈ 𝒫 𝑉 ∧ (𝐼𝑋) ≠ ∅))
4424, 42, 43sylanbrc 699 . 2 ((𝜑 ∧ (𝑉 ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 𝑉)) → (𝐼𝑋) ∈ (𝒫 𝑉 ∖ {∅}))
458, 44mpdan 703 1 (𝜑 → (𝐼𝑋) ∈ (𝒫 𝑉 ∖ {∅}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  cdif 3604  wss 3607  c0 3948  𝒫 cpw 4191  {csn 4210   class class class wbr 4685  dom cdm 5143  ran crn 5144  Fun wfun 5920   Fn wfn 5921  cfv 5926  Vtxcvtx 25919  iEdgciedg 25920  Edgcedg 25984  UHGraphcuhgr 25996   SubGraph csubgr 26204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-edg 25985  df-uhgr 25998  df-subgr 26205
This theorem is referenced by:  subumgredg2  26222  subuhgr  26223  subupgr  26224
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