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Theorem uhgrspansubgrlem 26075
Description: Lemma for uhgrspansubgr 26076: 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 26076. (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 edgval 25841 . . . . 5 (𝑆𝑊 → (Edg‘𝑆) = ran (iEdg‘𝑆))
31, 2syl 17 . . . 4 (𝜑 → (Edg‘𝑆) = ran (iEdg‘𝑆))
43eleq2d 2684 . . 3 (𝜑 → (𝑒 ∈ (Edg‘𝑆) ↔ 𝑒 ∈ ran (iEdg‘𝑆)))
5 uhgrspan.g . . . . . . . 8 (𝜑𝐺 ∈ UHGraph )
6 uhgrspan.e . . . . . . . . 9 𝐸 = (iEdg‘𝐺)
76uhgrfun 25857 . . . . . . . 8 (𝐺 ∈ UHGraph → Fun 𝐸)
85, 7syl 17 . . . . . . 7 (𝜑 → Fun 𝐸)
9 funres 5887 . . . . . . 7 (Fun 𝐸 → Fun (𝐸𝐴))
108, 9syl 17 . . . . . 6 (𝜑 → Fun (𝐸𝐴))
11 uhgrspan.r . . . . . . 7 (𝜑 → (iEdg‘𝑆) = (𝐸𝐴))
1211funeqd 5869 . . . . . 6 (𝜑 → (Fun (iEdg‘𝑆) ↔ Fun (𝐸𝐴)))
1310, 12mpbird 247 . . . . 5 (𝜑 → Fun (iEdg‘𝑆))
14 elrnrexdmb 6320 . . . . 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 481 . . . . . . . . 9 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → (iEdg‘𝑆) = (𝐸𝐴))
1716fveq1d 6150 . . . . . . . 8 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑖) = ((𝐸𝐴)‘𝑖))
1811dmeqd 5286 . . . . . . . . . . . . 13 (𝜑 → dom (iEdg‘𝑆) = dom (𝐸𝐴))
19 dmres 5378 . . . . . . . . . . . . 13 dom (𝐸𝐴) = (𝐴 ∩ dom 𝐸)
2018, 19syl6eq 2671 . . . . . . . . . . . 12 (𝜑 → dom (iEdg‘𝑆) = (𝐴 ∩ dom 𝐸))
2120eleq2d 2684 . . . . . . . . . . 11 (𝜑 → (𝑖 ∈ dom (iEdg‘𝑆) ↔ 𝑖 ∈ (𝐴 ∩ dom 𝐸)))
22 elinel1 3777 . . . . . . . . . . 11 (𝑖 ∈ (𝐴 ∩ dom 𝐸) → 𝑖𝐴)
2321, 22syl6bi 243 . . . . . . . . . 10 (𝜑 → (𝑖 ∈ dom (iEdg‘𝑆) → 𝑖𝐴))
2423imp 445 . . . . . . . . 9 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → 𝑖𝐴)
25 fvres 6164 . . . . . . . . 9 (𝑖𝐴 → ((𝐸𝐴)‘𝑖) = (𝐸𝑖))
2624, 25syl 17 . . . . . . . 8 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → ((𝐸𝐴)‘𝑖) = (𝐸𝑖))
2717, 26eqtrd 2655 . . . . . . 7 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑖) = (𝐸𝑖))
285adantr 481 . . . . . . . . 9 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → 𝐺 ∈ UHGraph )
29 elinel2 3778 . . . . . . . . . . 11 (𝑖 ∈ (𝐴 ∩ dom 𝐸) → 𝑖 ∈ dom 𝐸)
3021, 29syl6bi 243 . . . . . . . . . 10 (𝜑 → (𝑖 ∈ dom (iEdg‘𝑆) → 𝑖 ∈ dom 𝐸))
3130imp 445 . . . . . . . . 9 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → 𝑖 ∈ dom 𝐸)
32 uhgrspan.v . . . . . . . . . 10 𝑉 = (Vtx‘𝐺)
3332, 6uhgrss 25855 . . . . . . . . 9 ((𝐺 ∈ UHGraph ∧ 𝑖 ∈ dom 𝐸) → (𝐸𝑖) ⊆ 𝑉)
3428, 31, 33syl2anc 692 . . . . . . . 8 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → (𝐸𝑖) ⊆ 𝑉)
35 uhgrspan.q . . . . . . . . . . . 12 (𝜑 → (Vtx‘𝑆) = 𝑉)
3635pweqd 4135 . . . . . . . . . . 11 (𝜑 → 𝒫 (Vtx‘𝑆) = 𝒫 𝑉)
3736eleq2d 2684 . . . . . . . . . 10 (𝜑 → ((𝐸𝑖) ∈ 𝒫 (Vtx‘𝑆) ↔ (𝐸𝑖) ∈ 𝒫 𝑉))
3837adantr 481 . . . . . . . . 9 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → ((𝐸𝑖) ∈ 𝒫 (Vtx‘𝑆) ↔ (𝐸𝑖) ∈ 𝒫 𝑉))
39 fvex 6158 . . . . . . . . . 10 (𝐸𝑖) ∈ V
4039elpw 4136 . . . . . . . . 9 ((𝐸𝑖) ∈ 𝒫 𝑉 ↔ (𝐸𝑖) ⊆ 𝑉)
4138, 40syl6bb 276 . . . . . . . 8 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → ((𝐸𝑖) ∈ 𝒫 (Vtx‘𝑆) ↔ (𝐸𝑖) ⊆ 𝑉))
4234, 41mpbird 247 . . . . . . 7 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → (𝐸𝑖) ∈ 𝒫 (Vtx‘𝑆))
4327, 42eqeltrd 2698 . . . . . 6 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑖) ∈ 𝒫 (Vtx‘𝑆))
44 eleq1 2686 . . . . . 6 (𝑒 = ((iEdg‘𝑆)‘𝑖) → (𝑒 ∈ 𝒫 (Vtx‘𝑆) ↔ ((iEdg‘𝑆)‘𝑖) ∈ 𝒫 (Vtx‘𝑆)))
4543, 44syl5ibrcom 237 . . . . 5 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → (𝑒 = ((iEdg‘𝑆)‘𝑖) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
4645rexlimdva 3024 . . . 4 (𝜑 → (∃𝑖 ∈ dom (iEdg‘𝑆)𝑒 = ((iEdg‘𝑆)‘𝑖) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
4715, 46sylbid 230 . . 3 (𝜑 → (𝑒 ∈ ran (iEdg‘𝑆) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
484, 47sylbid 230 . 2 (𝜑 → (𝑒 ∈ (Edg‘𝑆) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
4948ssrdv 3589 1 (𝜑 → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wrex 2908  cin 3554  wss 3555  𝒫 cpw 4130  dom cdm 5074  ran crn 5075  cres 5076  Fun wfun 5841  cfv 5847  Vtxcvtx 25774  iEdgciedg 25775  Edgcedg 25839   UHGraph cuhgr 25847
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-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
This theorem is referenced by:  uhgrspansubgr  26076
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