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Theorem uhgrspansubgrlem 26586
Description: Lemma for uhgrspansubgr 26587: 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 26587. (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 edgval 26346 . . . 4 (Edg‘𝑆) = ran (iEdg‘𝑆)
21eleq2i 2897 . . 3 (𝑒 ∈ (Edg‘𝑆) ↔ 𝑒 ∈ ran (iEdg‘𝑆))
3 uhgrspan.g . . . . . . 7 (𝜑𝐺 ∈ UHGraph)
4 uhgrspan.e . . . . . . . 8 𝐸 = (iEdg‘𝐺)
54uhgrfun 26363 . . . . . . 7 (𝐺 ∈ UHGraph → Fun 𝐸)
6 funres 6164 . . . . . . 7 (Fun 𝐸 → Fun (𝐸𝐴))
73, 5, 63syl 18 . . . . . 6 (𝜑 → Fun (𝐸𝐴))
8 uhgrspan.r . . . . . . 7 (𝜑 → (iEdg‘𝑆) = (𝐸𝐴))
98funeqd 6144 . . . . . 6 (𝜑 → (Fun (iEdg‘𝑆) ↔ Fun (𝐸𝐴)))
107, 9mpbird 249 . . . . 5 (𝜑 → Fun (iEdg‘𝑆))
11 elrnrexdmb 6612 . . . . 5 (Fun (iEdg‘𝑆) → (𝑒 ∈ ran (iEdg‘𝑆) ↔ ∃𝑖 ∈ dom (iEdg‘𝑆)𝑒 = ((iEdg‘𝑆)‘𝑖)))
1210, 11syl 17 . . . 4 (𝜑 → (𝑒 ∈ ran (iEdg‘𝑆) ↔ ∃𝑖 ∈ dom (iEdg‘𝑆)𝑒 = ((iEdg‘𝑆)‘𝑖)))
138adantr 474 . . . . . . . . 9 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → (iEdg‘𝑆) = (𝐸𝐴))
1413fveq1d 6434 . . . . . . . 8 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑖) = ((𝐸𝐴)‘𝑖))
158dmeqd 5557 . . . . . . . . . . . . 13 (𝜑 → dom (iEdg‘𝑆) = dom (𝐸𝐴))
16 dmres 5654 . . . . . . . . . . . . 13 dom (𝐸𝐴) = (𝐴 ∩ dom 𝐸)
1715, 16syl6eq 2876 . . . . . . . . . . . 12 (𝜑 → dom (iEdg‘𝑆) = (𝐴 ∩ dom 𝐸))
1817eleq2d 2891 . . . . . . . . . . 11 (𝜑 → (𝑖 ∈ dom (iEdg‘𝑆) ↔ 𝑖 ∈ (𝐴 ∩ dom 𝐸)))
19 elinel1 4025 . . . . . . . . . . 11 (𝑖 ∈ (𝐴 ∩ dom 𝐸) → 𝑖𝐴)
2018, 19syl6bi 245 . . . . . . . . . 10 (𝜑 → (𝑖 ∈ dom (iEdg‘𝑆) → 𝑖𝐴))
2120imp 397 . . . . . . . . 9 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → 𝑖𝐴)
2221fvresd 6452 . . . . . . . 8 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → ((𝐸𝐴)‘𝑖) = (𝐸𝑖))
2314, 22eqtrd 2860 . . . . . . 7 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑖) = (𝐸𝑖))
24 elinel2 4026 . . . . . . . . . . 11 (𝑖 ∈ (𝐴 ∩ dom 𝐸) → 𝑖 ∈ dom 𝐸)
2518, 24syl6bi 245 . . . . . . . . . 10 (𝜑 → (𝑖 ∈ dom (iEdg‘𝑆) → 𝑖 ∈ dom 𝐸))
2625imp 397 . . . . . . . . 9 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → 𝑖 ∈ dom 𝐸)
27 uhgrspan.v . . . . . . . . . 10 𝑉 = (Vtx‘𝐺)
2827, 4uhgrss 26361 . . . . . . . . 9 ((𝐺 ∈ UHGraph ∧ 𝑖 ∈ dom 𝐸) → (𝐸𝑖) ⊆ 𝑉)
293, 26, 28syl2an2r 677 . . . . . . . 8 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → (𝐸𝑖) ⊆ 𝑉)
30 uhgrspan.q . . . . . . . . . . . 12 (𝜑 → (Vtx‘𝑆) = 𝑉)
3130pweqd 4382 . . . . . . . . . . 11 (𝜑 → 𝒫 (Vtx‘𝑆) = 𝒫 𝑉)
3231eleq2d 2891 . . . . . . . . . 10 (𝜑 → ((𝐸𝑖) ∈ 𝒫 (Vtx‘𝑆) ↔ (𝐸𝑖) ∈ 𝒫 𝑉))
3332adantr 474 . . . . . . . . 9 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → ((𝐸𝑖) ∈ 𝒫 (Vtx‘𝑆) ↔ (𝐸𝑖) ∈ 𝒫 𝑉))
34 fvex 6445 . . . . . . . . . 10 (𝐸𝑖) ∈ V
3534elpw 4383 . . . . . . . . 9 ((𝐸𝑖) ∈ 𝒫 𝑉 ↔ (𝐸𝑖) ⊆ 𝑉)
3633, 35syl6bb 279 . . . . . . . 8 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → ((𝐸𝑖) ∈ 𝒫 (Vtx‘𝑆) ↔ (𝐸𝑖) ⊆ 𝑉))
3729, 36mpbird 249 . . . . . . 7 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → (𝐸𝑖) ∈ 𝒫 (Vtx‘𝑆))
3823, 37eqeltrd 2905 . . . . . 6 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑖) ∈ 𝒫 (Vtx‘𝑆))
39 eleq1 2893 . . . . . 6 (𝑒 = ((iEdg‘𝑆)‘𝑖) → (𝑒 ∈ 𝒫 (Vtx‘𝑆) ↔ ((iEdg‘𝑆)‘𝑖) ∈ 𝒫 (Vtx‘𝑆)))
4038, 39syl5ibrcom 239 . . . . 5 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → (𝑒 = ((iEdg‘𝑆)‘𝑖) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
4140rexlimdva 3239 . . . 4 (𝜑 → (∃𝑖 ∈ dom (iEdg‘𝑆)𝑒 = ((iEdg‘𝑆)‘𝑖) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
4212, 41sylbid 232 . . 3 (𝜑 → (𝑒 ∈ ran (iEdg‘𝑆) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
432, 42syl5bi 234 . 2 (𝜑 → (𝑒 ∈ (Edg‘𝑆) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
4443ssrdv 3832 1 (𝜑 → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1658  wcel 2166  wrex 3117  cin 3796  wss 3797  𝒫 cpw 4377  dom cdm 5341  ran crn 5342  cres 5343  Fun wfun 6116  cfv 6122  Vtxcvtx 26293  iEdgciedg 26294  Edgcedg 26344  UHGraphcuhgr 26353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126  ax-un 7208
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-sbc 3662  df-csb 3757  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-pw 4379  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-br 4873  df-opab 4935  df-mpt 4952  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-fv 6130  df-edg 26345  df-uhgr 26355
This theorem is referenced by:  uhgrspansubgr  26587
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