MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  upgrreslem Structured version   Visualization version   GIF version

Theorem upgrreslem 28561
Description: Lemma for upgrres 28563. (Contributed by AV, 27-Nov-2020.) (Revised by AV, 19-Dec-2021.)
Hypotheses
Ref Expression
upgrres.v 𝑉 = (Vtx‘𝐺)
upgrres.e 𝐸 = (iEdg‘𝐺)
upgrres.f 𝐹 = {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)}
Assertion
Ref Expression
upgrreslem ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ran (𝐸𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
Distinct variable groups:   𝑖,𝐸   𝐸,𝑝   𝐺,𝑝   𝑖,𝑁   𝑁,𝑝   𝑉,𝑝
Allowed substitution hints:   𝐹(𝑖,𝑝)   𝐺(𝑖)   𝑉(𝑖)

Proof of Theorem upgrreslem
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 df-ima 5690 . 2 (𝐸𝐹) = ran (𝐸𝐹)
2 fveq2 6892 . . . . . . 7 (𝑖 = 𝑗 → (𝐸𝑖) = (𝐸𝑗))
3 neleq2 3054 . . . . . . 7 ((𝐸𝑖) = (𝐸𝑗) → (𝑁 ∉ (𝐸𝑖) ↔ 𝑁 ∉ (𝐸𝑗)))
42, 3syl 17 . . . . . 6 (𝑖 = 𝑗 → (𝑁 ∉ (𝐸𝑖) ↔ 𝑁 ∉ (𝐸𝑗)))
5 upgrres.f . . . . . 6 𝐹 = {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)}
64, 5elrab2 3687 . . . . 5 (𝑗𝐹 ↔ (𝑗 ∈ dom 𝐸𝑁 ∉ (𝐸𝑗)))
7 upgrres.v . . . . . . . 8 𝑉 = (Vtx‘𝐺)
8 upgrres.e . . . . . . . 8 𝐸 = (iEdg‘𝐺)
97, 8upgrf 28346 . . . . . . 7 (𝐺 ∈ UPGraph → 𝐸:dom 𝐸⟶{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
10 ffvelcdm 7084 . . . . . . . . . 10 ((𝐸:dom 𝐸⟶{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} ∧ 𝑗 ∈ dom 𝐸) → (𝐸𝑗) ∈ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
11 fveq2 6892 . . . . . . . . . . . . 13 (𝑝 = (𝐸𝑗) → (♯‘𝑝) = (♯‘(𝐸𝑗)))
1211breq1d 5159 . . . . . . . . . . . 12 (𝑝 = (𝐸𝑗) → ((♯‘𝑝) ≤ 2 ↔ (♯‘(𝐸𝑗)) ≤ 2))
1312elrab 3684 . . . . . . . . . . 11 ((𝐸𝑗) ∈ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} ↔ ((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2))
14 eldifsn 4791 . . . . . . . . . . . . . . . . . 18 ((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ↔ ((𝐸𝑗) ∈ 𝒫 𝑉 ∧ (𝐸𝑗) ≠ ∅))
15 simpl 484 . . . . . . . . . . . . . . . . . . . . 21 (((𝐸𝑗) ∈ 𝒫 𝑉𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ∈ 𝒫 𝑉)
16 elpwi 4610 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐸𝑗) ∈ 𝒫 𝑉 → (𝐸𝑗) ⊆ 𝑉)
1716adantr 482 . . . . . . . . . . . . . . . . . . . . 21 (((𝐸𝑗) ∈ 𝒫 𝑉𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ⊆ 𝑉)
18 simpr 486 . . . . . . . . . . . . . . . . . . . . 21 (((𝐸𝑗) ∈ 𝒫 𝑉𝑁 ∉ (𝐸𝑗)) → 𝑁 ∉ (𝐸𝑗))
19 elpwdifsn 4793 . . . . . . . . . . . . . . . . . . . . 21 (((𝐸𝑗) ∈ 𝒫 𝑉 ∧ (𝐸𝑗) ⊆ 𝑉𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
2015, 17, 18, 19syl3anc 1372 . . . . . . . . . . . . . . . . . . . 20 (((𝐸𝑗) ∈ 𝒫 𝑉𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
2120ex 414 . . . . . . . . . . . . . . . . . . 19 ((𝐸𝑗) ∈ 𝒫 𝑉 → (𝑁 ∉ (𝐸𝑗) → (𝐸𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
2221adantr 482 . . . . . . . . . . . . . . . . . 18 (((𝐸𝑗) ∈ 𝒫 𝑉 ∧ (𝐸𝑗) ≠ ∅) → (𝑁 ∉ (𝐸𝑗) → (𝐸𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
2314, 22sylbi 216 . . . . . . . . . . . . . . . . 17 ((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) → (𝑁 ∉ (𝐸𝑗) → (𝐸𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
2423adantr 482 . . . . . . . . . . . . . . . 16 (((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2) → (𝑁 ∉ (𝐸𝑗) → (𝐸𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
2524imp 408 . . . . . . . . . . . . . . 15 ((((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2) ∧ 𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
26 eldifsni 4794 . . . . . . . . . . . . . . . . 17 ((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) → (𝐸𝑗) ≠ ∅)
2726adantr 482 . . . . . . . . . . . . . . . 16 (((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2) → (𝐸𝑗) ≠ ∅)
2827adantr 482 . . . . . . . . . . . . . . 15 ((((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2) ∧ 𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ≠ ∅)
29 eldifsn 4791 . . . . . . . . . . . . . . 15 ((𝐸𝑗) ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ↔ ((𝐸𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}) ∧ (𝐸𝑗) ≠ ∅))
3025, 28, 29sylanbrc 584 . . . . . . . . . . . . . 14 ((((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2) ∧ 𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}))
31 simpr 486 . . . . . . . . . . . . . . 15 (((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2) → (♯‘(𝐸𝑗)) ≤ 2)
3231adantr 482 . . . . . . . . . . . . . 14 ((((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2) ∧ 𝑁 ∉ (𝐸𝑗)) → (♯‘(𝐸𝑗)) ≤ 2)
3312, 30, 32elrabd 3686 . . . . . . . . . . . . 13 ((((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2) ∧ 𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
3433ex 414 . . . . . . . . . . . 12 (((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2) → (𝑁 ∉ (𝐸𝑗) → (𝐸𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
3534a1d 25 . . . . . . . . . . 11 (((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2) → (𝑁𝑉 → (𝑁 ∉ (𝐸𝑗) → (𝐸𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})))
3613, 35sylbi 216 . . . . . . . . . 10 ((𝐸𝑗) ∈ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} → (𝑁𝑉 → (𝑁 ∉ (𝐸𝑗) → (𝐸𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})))
3710, 36syl 17 . . . . . . . . 9 ((𝐸:dom 𝐸⟶{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} ∧ 𝑗 ∈ dom 𝐸) → (𝑁𝑉 → (𝑁 ∉ (𝐸𝑗) → (𝐸𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})))
3837ex 414 . . . . . . . 8 (𝐸:dom 𝐸⟶{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} → (𝑗 ∈ dom 𝐸 → (𝑁𝑉 → (𝑁 ∉ (𝐸𝑗) → (𝐸𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))))
3938com23 86 . . . . . . 7 (𝐸:dom 𝐸⟶{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} → (𝑁𝑉 → (𝑗 ∈ dom 𝐸 → (𝑁 ∉ (𝐸𝑗) → (𝐸𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))))
409, 39syl 17 . . . . . 6 (𝐺 ∈ UPGraph → (𝑁𝑉 → (𝑗 ∈ dom 𝐸 → (𝑁 ∉ (𝐸𝑗) → (𝐸𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))))
4140imp4b 423 . . . . 5 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ((𝑗 ∈ dom 𝐸𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
426, 41biimtrid 241 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → (𝑗𝐹 → (𝐸𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
4342ralrimiv 3146 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ∀𝑗𝐹 (𝐸𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
44 upgruhgr 28362 . . . . . 6 (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
458uhgrfun 28326 . . . . . 6 (𝐺 ∈ UHGraph → Fun 𝐸)
4644, 45syl 17 . . . . 5 (𝐺 ∈ UPGraph → Fun 𝐸)
4746adantr 482 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → Fun 𝐸)
485ssrab3 4081 . . . 4 𝐹 ⊆ dom 𝐸
49 funimass4 6957 . . . 4 ((Fun 𝐸𝐹 ⊆ dom 𝐸) → ((𝐸𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} ↔ ∀𝑗𝐹 (𝐸𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
5047, 48, 49sylancl 587 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ((𝐸𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} ↔ ∀𝑗𝐹 (𝐸𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
5143, 50mpbird 257 . 2 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → (𝐸𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
521, 51eqsstrrid 4032 1 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ran (𝐸𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wne 2941  wnel 3047  wral 3062  {crab 3433  cdif 3946  wss 3949  c0 4323  𝒫 cpw 4603  {csn 4629   class class class wbr 5149  dom cdm 5677  ran crn 5678  cres 5679  cima 5680  Fun wfun 6538  wf 6540  cfv 6544  cle 11249  2c2 12267  chash 14290  Vtxcvtx 28256  iEdgciedg 28257  UHGraphcuhgr 28316  UPGraphcupgr 28340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-uhgr 28318  df-upgr 28342
This theorem is referenced by:  upgrres  28563
  Copyright terms: Public domain W3C validator