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Theorem uhgrspansubgrlem 27058
Description: Lemma for uhgrspansubgr 27059: 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 27059. (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 26820 . . . 4 (Edg‘𝑆) = ran (iEdg‘𝑆)
21eleq2i 2903 . . 3 (𝑒 ∈ (Edg‘𝑆) ↔ 𝑒 ∈ ran (iEdg‘𝑆))
3 uhgrspan.g . . . . . . 7 (𝜑𝐺 ∈ UHGraph)
4 uhgrspan.e . . . . . . . 8 𝐸 = (iEdg‘𝐺)
54uhgrfun 26837 . . . . . . 7 (𝐺 ∈ UHGraph → Fun 𝐸)
6 funres 6370 . . . . . . 7 (Fun 𝐸 → Fun (𝐸𝐴))
73, 5, 63syl 18 . . . . . 6 (𝜑 → Fun (𝐸𝐴))
8 uhgrspan.r . . . . . . 7 (𝜑 → (iEdg‘𝑆) = (𝐸𝐴))
98funeqd 6350 . . . . . 6 (𝜑 → (Fun (iEdg‘𝑆) ↔ Fun (𝐸𝐴)))
107, 9mpbird 260 . . . . 5 (𝜑 → Fun (iEdg‘𝑆))
11 elrnrexdmb 6829 . . . . 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 484 . . . . . . . . 9 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → (iEdg‘𝑆) = (𝐸𝐴))
1413fveq1d 6645 . . . . . . . 8 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑖) = ((𝐸𝐴)‘𝑖))
158dmeqd 5747 . . . . . . . . . . . . 13 (𝜑 → dom (iEdg‘𝑆) = dom (𝐸𝐴))
16 dmres 5848 . . . . . . . . . . . . 13 dom (𝐸𝐴) = (𝐴 ∩ dom 𝐸)
1715, 16syl6eq 2872 . . . . . . . . . . . 12 (𝜑 → dom (iEdg‘𝑆) = (𝐴 ∩ dom 𝐸))
1817eleq2d 2897 . . . . . . . . . . 11 (𝜑 → (𝑖 ∈ dom (iEdg‘𝑆) ↔ 𝑖 ∈ (𝐴 ∩ dom 𝐸)))
19 elinel1 4147 . . . . . . . . . . 11 (𝑖 ∈ (𝐴 ∩ dom 𝐸) → 𝑖𝐴)
2018, 19syl6bi 256 . . . . . . . . . 10 (𝜑 → (𝑖 ∈ dom (iEdg‘𝑆) → 𝑖𝐴))
2120imp 410 . . . . . . . . 9 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → 𝑖𝐴)
2221fvresd 6663 . . . . . . . 8 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → ((𝐸𝐴)‘𝑖) = (𝐸𝑖))
2314, 22eqtrd 2856 . . . . . . 7 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑖) = (𝐸𝑖))
24 elinel2 4148 . . . . . . . . . . 11 (𝑖 ∈ (𝐴 ∩ dom 𝐸) → 𝑖 ∈ dom 𝐸)
2518, 24syl6bi 256 . . . . . . . . . 10 (𝜑 → (𝑖 ∈ dom (iEdg‘𝑆) → 𝑖 ∈ dom 𝐸))
2625imp 410 . . . . . . . . 9 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → 𝑖 ∈ dom 𝐸)
27 uhgrspan.v . . . . . . . . . 10 𝑉 = (Vtx‘𝐺)
2827, 4uhgrss 26835 . . . . . . . . 9 ((𝐺 ∈ UHGraph ∧ 𝑖 ∈ dom 𝐸) → (𝐸𝑖) ⊆ 𝑉)
293, 26, 28syl2an2r 684 . . . . . . . 8 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → (𝐸𝑖) ⊆ 𝑉)
30 uhgrspan.q . . . . . . . . . . . 12 (𝜑 → (Vtx‘𝑆) = 𝑉)
3130pweqd 4531 . . . . . . . . . . 11 (𝜑 → 𝒫 (Vtx‘𝑆) = 𝒫 𝑉)
3231eleq2d 2897 . . . . . . . . . 10 (𝜑 → ((𝐸𝑖) ∈ 𝒫 (Vtx‘𝑆) ↔ (𝐸𝑖) ∈ 𝒫 𝑉))
3332adantr 484 . . . . . . . . 9 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → ((𝐸𝑖) ∈ 𝒫 (Vtx‘𝑆) ↔ (𝐸𝑖) ∈ 𝒫 𝑉))
34 fvex 6656 . . . . . . . . . 10 (𝐸𝑖) ∈ V
3534elpw 4516 . . . . . . . . 9 ((𝐸𝑖) ∈ 𝒫 𝑉 ↔ (𝐸𝑖) ⊆ 𝑉)
3633, 35syl6bb 290 . . . . . . . 8 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → ((𝐸𝑖) ∈ 𝒫 (Vtx‘𝑆) ↔ (𝐸𝑖) ⊆ 𝑉))
3729, 36mpbird 260 . . . . . . 7 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → (𝐸𝑖) ∈ 𝒫 (Vtx‘𝑆))
3823, 37eqeltrd 2912 . . . . . 6 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑖) ∈ 𝒫 (Vtx‘𝑆))
39 eleq1 2899 . . . . . 6 (𝑒 = ((iEdg‘𝑆)‘𝑖) → (𝑒 ∈ 𝒫 (Vtx‘𝑆) ↔ ((iEdg‘𝑆)‘𝑖) ∈ 𝒫 (Vtx‘𝑆)))
4038, 39syl5ibrcom 250 . . . . 5 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → (𝑒 = ((iEdg‘𝑆)‘𝑖) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
4140rexlimdva 3270 . . . 4 (𝜑 → (∃𝑖 ∈ dom (iEdg‘𝑆)𝑒 = ((iEdg‘𝑆)‘𝑖) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
4212, 41sylbid 243 . . 3 (𝜑 → (𝑒 ∈ ran (iEdg‘𝑆) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
432, 42syl5bi 245 . 2 (𝜑 → (𝑒 ∈ (Edg‘𝑆) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
4443ssrdv 3949 1 (𝜑 → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wrex 3127  cin 3909  wss 3910  𝒫 cpw 4512  dom cdm 5528  ran crn 5529  cres 5530  Fun wfun 6322  cfv 6328  Vtxcvtx 26767  iEdgciedg 26768  Edgcedg 26818  UHGraphcuhgr 26827
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
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 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-edg 26819  df-uhgr 26829
This theorem is referenced by:  uhgrspansubgr  27059
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