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

Theorem upgrreslem 29321
Description: Lemma for upgrres 29323. (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 5698 . 2 (𝐸𝐹) = ran (𝐸𝐹)
2 fveq2 6906 . . . . . . 7 (𝑖 = 𝑗 → (𝐸𝑖) = (𝐸𝑗))
3 neleq2 3053 . . . . . . 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 3695 . . . . 5 (𝑗𝐹 ↔ (𝑗 ∈ dom 𝐸𝑁 ∉ (𝐸𝑗)))
7 upgrres.v . . . . . . . 8 𝑉 = (Vtx‘𝐺)
8 upgrres.e . . . . . . . 8 𝐸 = (iEdg‘𝐺)
97, 8upgrf 29103 . . . . . . 7 (𝐺 ∈ UPGraph → 𝐸:dom 𝐸⟶{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
10 ffvelcdm 7101 . . . . . . . . . 10 ((𝐸:dom 𝐸⟶{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} ∧ 𝑗 ∈ dom 𝐸) → (𝐸𝑗) ∈ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
11 fveq2 6906 . . . . . . . . . . . . 13 (𝑝 = (𝐸𝑗) → (♯‘𝑝) = (♯‘(𝐸𝑗)))
1211breq1d 5153 . . . . . . . . . . . 12 (𝑝 = (𝐸𝑗) → ((♯‘𝑝) ≤ 2 ↔ (♯‘(𝐸𝑗)) ≤ 2))
1312elrab 3692 . . . . . . . . . . 11 ((𝐸𝑗) ∈ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} ↔ ((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2))
14 eldifsn 4786 . . . . . . . . . . . . . . . . . 18 ((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ↔ ((𝐸𝑗) ∈ 𝒫 𝑉 ∧ (𝐸𝑗) ≠ ∅))
15 simpl 482 . . . . . . . . . . . . . . . . . . . . 21 (((𝐸𝑗) ∈ 𝒫 𝑉𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ∈ 𝒫 𝑉)
16 elpwi 4607 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐸𝑗) ∈ 𝒫 𝑉 → (𝐸𝑗) ⊆ 𝑉)
1716adantr 480 . . . . . . . . . . . . . . . . . . . . 21 (((𝐸𝑗) ∈ 𝒫 𝑉𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ⊆ 𝑉)
18 simpr 484 . . . . . . . . . . . . . . . . . . . . 21 (((𝐸𝑗) ∈ 𝒫 𝑉𝑁 ∉ (𝐸𝑗)) → 𝑁 ∉ (𝐸𝑗))
19 elpwdifsn 4789 . . . . . . . . . . . . . . . . . . . . 21 (((𝐸𝑗) ∈ 𝒫 𝑉 ∧ (𝐸𝑗) ⊆ 𝑉𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
2015, 17, 18, 19syl3anc 1373 . . . . . . . . . . . . . . . . . . . 20 (((𝐸𝑗) ∈ 𝒫 𝑉𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
2120ex 412 . . . . . . . . . . . . . . . . . . 19 ((𝐸𝑗) ∈ 𝒫 𝑉 → (𝑁 ∉ (𝐸𝑗) → (𝐸𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
2221adantr 480 . . . . . . . . . . . . . . . . . 18 (((𝐸𝑗) ∈ 𝒫 𝑉 ∧ (𝐸𝑗) ≠ ∅) → (𝑁 ∉ (𝐸𝑗) → (𝐸𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
2314, 22sylbi 217 . . . . . . . . . . . . . . . . 17 ((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) → (𝑁 ∉ (𝐸𝑗) → (𝐸𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
2423adantr 480 . . . . . . . . . . . . . . . 16 (((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2) → (𝑁 ∉ (𝐸𝑗) → (𝐸𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
2524imp 406 . . . . . . . . . . . . . . 15 ((((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2) ∧ 𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
26 eldifsni 4790 . . . . . . . . . . . . . . . . 17 ((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) → (𝐸𝑗) ≠ ∅)
2726adantr 480 . . . . . . . . . . . . . . . 16 (((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2) → (𝐸𝑗) ≠ ∅)
2827adantr 480 . . . . . . . . . . . . . . 15 ((((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2) ∧ 𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ≠ ∅)
29 eldifsn 4786 . . . . . . . . . . . . . . 15 ((𝐸𝑗) ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ↔ ((𝐸𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}) ∧ (𝐸𝑗) ≠ ∅))
3025, 28, 29sylanbrc 583 . . . . . . . . . . . . . 14 ((((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2) ∧ 𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}))
31 simpr 484 . . . . . . . . . . . . . . 15 (((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2) → (♯‘(𝐸𝑗)) ≤ 2)
3231adantr 480 . . . . . . . . . . . . . 14 ((((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2) ∧ 𝑁 ∉ (𝐸𝑗)) → (♯‘(𝐸𝑗)) ≤ 2)
3312, 30, 32elrabd 3694 . . . . . . . . . . . . 13 ((((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2) ∧ 𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
3433ex 412 . . . . . . . . . . . 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 217 . . . . . . . . . 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 412 . . . . . . . 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 421 . . . . 5 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ((𝑗 ∈ dom 𝐸𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
426, 41biimtrid 242 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → (𝑗𝐹 → (𝐸𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
4342ralrimiv 3145 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ∀𝑗𝐹 (𝐸𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
44 upgruhgr 29119 . . . . . 6 (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
458uhgrfun 29083 . . . . . 6 (𝐺 ∈ UHGraph → Fun 𝐸)
4644, 45syl 17 . . . . 5 (𝐺 ∈ UPGraph → Fun 𝐸)
4746adantr 480 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → Fun 𝐸)
485ssrab3 4082 . . . 4 𝐹 ⊆ dom 𝐸
49 funimass4 6973 . . . 4 ((Fun 𝐸𝐹 ⊆ dom 𝐸) → ((𝐸𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} ↔ ∀𝑗𝐹 (𝐸𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
5047, 48, 49sylancl 586 . . 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 4023 1 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ran (𝐸𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  wnel 3046  wral 3061  {crab 3436  cdif 3948  wss 3951  c0 4333  𝒫 cpw 4600  {csn 4626   class class class wbr 5143  dom cdm 5685  ran crn 5686  cres 5687  cima 5688  Fun wfun 6555  wf 6557  cfv 6561  cle 11296  2c2 12321  chash 14369  Vtxcvtx 29013  iEdgciedg 29014  UHGraphcuhgr 29073  UPGraphcupgr 29097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-uhgr 29075  df-upgr 29099
This theorem is referenced by:  upgrres  29323
  Copyright terms: Public domain W3C validator