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Theorem uhgrspansubgrlem 26405
Description: Lemma for uhgrspansubgr 26406: 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 26406. (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 26162 . . . 4 (Edg‘𝑆) = ran (iEdg‘𝑆)
21eleq2i 2842 . . 3 (𝑒 ∈ (Edg‘𝑆) ↔ 𝑒 ∈ ran (iEdg‘𝑆))
3 uhgrspan.g . . . . . . 7 (𝜑𝐺 ∈ UHGraph)
4 uhgrspan.e . . . . . . . 8 𝐸 = (iEdg‘𝐺)
54uhgrfun 26182 . . . . . . 7 (𝐺 ∈ UHGraph → Fun 𝐸)
6 funres 6072 . . . . . . 7 (Fun 𝐸 → Fun (𝐸𝐴))
73, 5, 63syl 18 . . . . . 6 (𝜑 → Fun (𝐸𝐴))
8 uhgrspan.r . . . . . . 7 (𝜑 → (iEdg‘𝑆) = (𝐸𝐴))
98funeqd 6053 . . . . . 6 (𝜑 → (Fun (iEdg‘𝑆) ↔ Fun (𝐸𝐴)))
107, 9mpbird 247 . . . . 5 (𝜑 → Fun (iEdg‘𝑆))
11 elrnrexdmb 6507 . . . . 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 466 . . . . . . . . 9 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → (iEdg‘𝑆) = (𝐸𝐴))
1413fveq1d 6334 . . . . . . . 8 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑖) = ((𝐸𝐴)‘𝑖))
158dmeqd 5464 . . . . . . . . . . . . 13 (𝜑 → dom (iEdg‘𝑆) = dom (𝐸𝐴))
16 dmres 5560 . . . . . . . . . . . . 13 dom (𝐸𝐴) = (𝐴 ∩ dom 𝐸)
1715, 16syl6eq 2821 . . . . . . . . . . . 12 (𝜑 → dom (iEdg‘𝑆) = (𝐴 ∩ dom 𝐸))
1817eleq2d 2836 . . . . . . . . . . 11 (𝜑 → (𝑖 ∈ dom (iEdg‘𝑆) ↔ 𝑖 ∈ (𝐴 ∩ dom 𝐸)))
19 elinel1 3950 . . . . . . . . . . 11 (𝑖 ∈ (𝐴 ∩ dom 𝐸) → 𝑖𝐴)
2018, 19syl6bi 243 . . . . . . . . . 10 (𝜑 → (𝑖 ∈ dom (iEdg‘𝑆) → 𝑖𝐴))
2120imp 393 . . . . . . . . 9 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → 𝑖𝐴)
2221fvresd 6349 . . . . . . . 8 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → ((𝐸𝐴)‘𝑖) = (𝐸𝑖))
2314, 22eqtrd 2805 . . . . . . 7 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑖) = (𝐸𝑖))
24 elinel2 3951 . . . . . . . . . . 11 (𝑖 ∈ (𝐴 ∩ dom 𝐸) → 𝑖 ∈ dom 𝐸)
2518, 24syl6bi 243 . . . . . . . . . 10 (𝜑 → (𝑖 ∈ dom (iEdg‘𝑆) → 𝑖 ∈ dom 𝐸))
2625imp 393 . . . . . . . . 9 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → 𝑖 ∈ dom 𝐸)
27 uhgrspan.v . . . . . . . . . 10 𝑉 = (Vtx‘𝐺)
2827, 4uhgrss 26180 . . . . . . . . 9 ((𝐺 ∈ UHGraph ∧ 𝑖 ∈ dom 𝐸) → (𝐸𝑖) ⊆ 𝑉)
293, 26, 28syl2an2r 664 . . . . . . . 8 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → (𝐸𝑖) ⊆ 𝑉)
30 uhgrspan.q . . . . . . . . . . . 12 (𝜑 → (Vtx‘𝑆) = 𝑉)
3130pweqd 4302 . . . . . . . . . . 11 (𝜑 → 𝒫 (Vtx‘𝑆) = 𝒫 𝑉)
3231eleq2d 2836 . . . . . . . . . 10 (𝜑 → ((𝐸𝑖) ∈ 𝒫 (Vtx‘𝑆) ↔ (𝐸𝑖) ∈ 𝒫 𝑉))
3332adantr 466 . . . . . . . . 9 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → ((𝐸𝑖) ∈ 𝒫 (Vtx‘𝑆) ↔ (𝐸𝑖) ∈ 𝒫 𝑉))
34 fvex 6342 . . . . . . . . . 10 (𝐸𝑖) ∈ V
3534elpw 4303 . . . . . . . . 9 ((𝐸𝑖) ∈ 𝒫 𝑉 ↔ (𝐸𝑖) ⊆ 𝑉)
3633, 35syl6bb 276 . . . . . . . 8 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → ((𝐸𝑖) ∈ 𝒫 (Vtx‘𝑆) ↔ (𝐸𝑖) ⊆ 𝑉))
3729, 36mpbird 247 . . . . . . 7 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → (𝐸𝑖) ∈ 𝒫 (Vtx‘𝑆))
3823, 37eqeltrd 2850 . . . . . 6 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑖) ∈ 𝒫 (Vtx‘𝑆))
39 eleq1 2838 . . . . . 6 (𝑒 = ((iEdg‘𝑆)‘𝑖) → (𝑒 ∈ 𝒫 (Vtx‘𝑆) ↔ ((iEdg‘𝑆)‘𝑖) ∈ 𝒫 (Vtx‘𝑆)))
4038, 39syl5ibrcom 237 . . . . 5 ((𝜑𝑖 ∈ dom (iEdg‘𝑆)) → (𝑒 = ((iEdg‘𝑆)‘𝑖) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
4140rexlimdva 3179 . . . 4 (𝜑 → (∃𝑖 ∈ dom (iEdg‘𝑆)𝑒 = ((iEdg‘𝑆)‘𝑖) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
4212, 41sylbid 230 . . 3 (𝜑 → (𝑒 ∈ ran (iEdg‘𝑆) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
432, 42syl5bi 232 . 2 (𝜑 → (𝑒 ∈ (Edg‘𝑆) → 𝑒 ∈ 𝒫 (Vtx‘𝑆)))
4443ssrdv 3758 1 (𝜑 → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wrex 3062  cin 3722  wss 3723  𝒫 cpw 4297  dom cdm 5249  ran crn 5250  cres 5251  Fun wfun 6025  cfv 6031  Vtxcvtx 26095  iEdgciedg 26096  Edgcedg 26160  UHGraphcuhgr 26172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-edg 26161  df-uhgr 26174
This theorem is referenced by:  uhgrspansubgr  26406
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