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Theorem uhgrspansubgrlem 40514
 Description: Lemma for uhgrspansubgr 40515: The edges of the graph 𝑆 obtained by removing some edges of a hypergraph 𝐺 are subsets of its vertices (a spanning subgraph, see comment for uhgrspansubgr 40515. (Contributed by AV, 18-Nov-2020.)
Hypotheses
Ref Expression
uhgrspan.v 𝑉 = (Vtx‘𝐺)
uhgrspan.e 𝐸 = (iEdg‘𝐺)
uhgrspan.s (𝜑𝑆𝑊)
uhgrspan.q (𝜑 → (Vtx‘𝑆) = 𝑉)
uhgrspan.r (𝜑 → (iEdg‘𝑆) = (𝐸𝐴))
uhgrspan.g (𝜑𝐺 ∈ UHGraph )
Assertion
Ref Expression
uhgrspansubgrlem (𝜑 → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))

Proof of Theorem uhgrspansubgrlem
Dummy variables 𝑒 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uhgrspan.s . . . . 5 (𝜑𝑆𝑊)
2 edgaval 25794 . . . . 5 (𝑆𝑊 → (Edg‘𝑆) = ran (iEdg‘𝑆))
31, 2syl 17 . . . 4 (𝜑 → (Edg‘𝑆) = ran (iEdg‘𝑆))
43eleq2d 2673 . . 3 (𝜑 → (𝑒 ∈ (Edg‘𝑆) ↔ 𝑒 ∈ ran (iEdg‘𝑆)))
5 uhgrspan.g . . . . . . . 8 (𝜑𝐺 ∈ UHGraph )
6 uhgrspan.e . . . . . . . . 9 𝐸 = (iEdg‘𝐺)
76uhgrfun 25732 . . . . . . . 8 (𝐺 ∈ UHGraph → Fun 𝐸)
85, 7syl 17 . . . . . . 7 (𝜑 → Fun 𝐸)
9 funres 5843 . . . . . . 7 (Fun 𝐸 → Fun (𝐸𝐴))
108, 9syl 17 . . . . . 6 (𝜑 → Fun (𝐸𝐴))
11 uhgrspan.r . . . . . . 7 (𝜑 → (iEdg‘𝑆) = (𝐸𝐴))
1211funeqd 5825 . . . . . 6 (𝜑 → (Fun (iEdg‘𝑆) ↔ Fun (𝐸𝐴)))
1310, 12mpbird 246 . . . . 5 (𝜑 → Fun (iEdg‘𝑆))
14 elrnrexdmb 6272 . . . . 5 (Fun (iEdg‘𝑆) → (𝑒 ∈ ran (iEdg‘𝑆) ↔ ∃𝑖 ∈ dom (iEdg‘𝑆)𝑒 = ((iEdg‘𝑆)‘𝑖)))
1513, 14syl 17 . . . 4 (𝜑 → (𝑒 ∈ ran (iEdg‘𝑆) ↔ ∃𝑖 ∈ dom (iEdg‘𝑆)𝑒 = ((iEdg‘𝑆)‘𝑖)))
1611adantr 480 . . . . . . . . 9 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → (iEdg‘𝑆) = (𝐸𝐴))
1716fveq1d 6105 . . . . . . . 8 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑖) = ((𝐸𝐴)‘𝑖))
1811dmeqd 5248 . . . . . . . . . . . . 13 (𝜑 → dom (iEdg‘𝑆) = dom (𝐸𝐴))
19 dmres 5339 . . . . . . . . . . . . 13 dom (𝐸𝐴) = (𝐴 ∩ dom 𝐸)
2018, 19syl6eq 2660 . . . . . . . . . . . 12 (𝜑 → dom (iEdg‘𝑆) = (𝐴 ∩ dom 𝐸))
2120eleq2d 2673 . . . . . . . . . . 11 (𝜑 → (𝑖 ∈ dom (iEdg‘𝑆) ↔ 𝑖 ∈ (𝐴 ∩ dom 𝐸)))
22 elinel1 3761 . . . . . . . . . . 11 (𝑖 ∈ (𝐴 ∩ dom 𝐸) → 𝑖𝐴)
2321, 22syl6bi 242 . . . . . . . . . 10 (𝜑 → (𝑖 ∈ dom (iEdg‘𝑆) → 𝑖𝐴))
2423imp 444 . . . . . . . . 9 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → 𝑖𝐴)
25 fvres 6117 . . . . . . . . 9 (𝑖𝐴 → ((𝐸𝐴)‘𝑖) = (𝐸𝑖))
2624, 25syl 17 . . . . . . . 8 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → ((𝐸𝐴)‘𝑖) = (𝐸𝑖))
2717, 26eqtrd 2644 . . . . . . 7 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑖) = (𝐸𝑖))
285adantr 480 . . . . . . . . 9 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → 𝐺 ∈ UHGraph )
29 elinel2 3762 . . . . . . . . . . 11 (𝑖 ∈ (𝐴 ∩ dom 𝐸) → 𝑖 ∈ dom 𝐸)
3021, 29syl6bi 242 . . . . . . . . . 10 (𝜑 → (𝑖 ∈ dom (iEdg‘𝑆) → 𝑖 ∈ dom 𝐸))
3130imp 444 . . . . . . . . 9 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → 𝑖 ∈ dom 𝐸)
32 uhgrspan.v . . . . . . . . . 10 𝑉 = (Vtx‘𝐺)
3332, 6uhgrss 25730 . . . . . . . . 9 ((𝐺 ∈ UHGraph ∧ 𝑖 ∈ dom 𝐸) → (𝐸𝑖) ⊆ 𝑉)
3428, 31, 33syl2anc 691 . . . . . . . 8 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → (𝐸𝑖) ⊆ 𝑉)
35 uhgrspan.q . . . . . . . . . . . 12 (𝜑 → (Vtx‘𝑆) = 𝑉)
3635pweqd 4113 . . . . . . . . . . 11 (𝜑 → 𝒫 (Vtx‘𝑆) = 𝒫 𝑉)
3736eleq2d 2673 . . . . . . . . . 10 (𝜑 → ((𝐸𝑖) ∈ 𝒫 (Vtx‘𝑆) ↔ (𝐸𝑖) ∈ 𝒫 𝑉))
3837adantr 480 . . . . . . . . 9 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → ((𝐸𝑖) ∈ 𝒫 (Vtx‘𝑆) ↔ (𝐸𝑖) ∈ 𝒫 𝑉))
39 fvex 6113 . . . . . . . . . 10 (𝐸𝑖) ∈ V
4039elpw 4114 . . . . . . . . 9 ((𝐸𝑖) ∈ 𝒫 𝑉 ↔ (𝐸𝑖) ⊆ 𝑉)
4138, 40syl6bb 275 . . . . . . . 8 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → ((𝐸𝑖) ∈ 𝒫 (Vtx‘𝑆) ↔ (𝐸𝑖) ⊆ 𝑉))
4234, 41mpbird 246 . . . . . . 7 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → (𝐸𝑖) ∈ 𝒫 (Vtx‘𝑆))
4327, 42eqeltrd 2688 . . . . . 6 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑖) ∈ 𝒫 (Vtx‘𝑆))
44 eleq1 2676 . . . . . 6 (𝑒 = ((iEdg‘𝑆)‘𝑖) → (𝑒 ∈ 𝒫 (Vtx‘𝑆) ↔ ((iEdg‘𝑆)‘𝑖) ∈ 𝒫 (Vtx‘𝑆)))
4543, 44syl5ibrcom 236 . . . . 5 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → (𝑒 = ((iEdg‘𝑆)‘𝑖) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
4645rexlimdva 3013 . . . 4 (𝜑 → (∃𝑖 ∈ dom (iEdg‘𝑆)𝑒 = ((iEdg‘𝑆)‘𝑖) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
4715, 46sylbid 229 . . 3 (𝜑 → (𝑒 ∈ ran (iEdg‘𝑆) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
484, 47sylbid 229 . 2 (𝜑 → (𝑒 ∈ (Edg‘𝑆) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
4948ssrdv 3574 1 (𝜑 → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  dom cdm 5038  ran crn 5039   ↾ cres 5040  Fun wfun 5798  ‘cfv 5804  Vtxcvtx 25673  iEdgciedg 25674   UHGraph cuhgr 25722  Edgcedga 25792 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-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 This theorem is referenced by:  uhgrspansubgr  40515
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