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Theorem upgrreslem 27392
Description: Lemma for upgrres 27394. (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 5564 . 2 (𝐸𝐹) = ran (𝐸𝐹)
2 fveq2 6717 . . . . . . 7 (𝑖 = 𝑗 → (𝐸𝑖) = (𝐸𝑗))
3 neleq2 3052 . . . . . . 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 3605 . . . . 5 (𝑗𝐹 ↔ (𝑗 ∈ dom 𝐸𝑁 ∉ (𝐸𝑗)))
7 upgrres.v . . . . . . . 8 𝑉 = (Vtx‘𝐺)
8 upgrres.e . . . . . . . 8 𝐸 = (iEdg‘𝐺)
97, 8upgrf 27177 . . . . . . 7 (𝐺 ∈ UPGraph → 𝐸:dom 𝐸⟶{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
10 ffvelrn 6902 . . . . . . . . . 10 ((𝐸:dom 𝐸⟶{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} ∧ 𝑗 ∈ dom 𝐸) → (𝐸𝑗) ∈ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
11 fveq2 6717 . . . . . . . . . . . . 13 (𝑝 = (𝐸𝑗) → (♯‘𝑝) = (♯‘(𝐸𝑗)))
1211breq1d 5063 . . . . . . . . . . . 12 (𝑝 = (𝐸𝑗) → ((♯‘𝑝) ≤ 2 ↔ (♯‘(𝐸𝑗)) ≤ 2))
1312elrab 3602 . . . . . . . . . . 11 ((𝐸𝑗) ∈ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} ↔ ((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2))
14 eldifsn 4700 . . . . . . . . . . . . . . . . . 18 ((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ↔ ((𝐸𝑗) ∈ 𝒫 𝑉 ∧ (𝐸𝑗) ≠ ∅))
15 simpl 486 . . . . . . . . . . . . . . . . . . . . 21 (((𝐸𝑗) ∈ 𝒫 𝑉𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ∈ 𝒫 𝑉)
16 elpwi 4522 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐸𝑗) ∈ 𝒫 𝑉 → (𝐸𝑗) ⊆ 𝑉)
1716adantr 484 . . . . . . . . . . . . . . . . . . . . 21 (((𝐸𝑗) ∈ 𝒫 𝑉𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ⊆ 𝑉)
18 simpr 488 . . . . . . . . . . . . . . . . . . . . 21 (((𝐸𝑗) ∈ 𝒫 𝑉𝑁 ∉ (𝐸𝑗)) → 𝑁 ∉ (𝐸𝑗))
19 elpwdifsn 4702 . . . . . . . . . . . . . . . . . . . . 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 416 . . . . . . . . . . . . . . . . . . 19 ((𝐸𝑗) ∈ 𝒫 𝑉 → (𝑁 ∉ (𝐸𝑗) → (𝐸𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
2221adantr 484 . . . . . . . . . . . . . . . . . 18 (((𝐸𝑗) ∈ 𝒫 𝑉 ∧ (𝐸𝑗) ≠ ∅) → (𝑁 ∉ (𝐸𝑗) → (𝐸𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
2314, 22sylbi 220 . . . . . . . . . . . . . . . . 17 ((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) → (𝑁 ∉ (𝐸𝑗) → (𝐸𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
2423adantr 484 . . . . . . . . . . . . . . . 16 (((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2) → (𝑁 ∉ (𝐸𝑗) → (𝐸𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
2524imp 410 . . . . . . . . . . . . . . 15 ((((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2) ∧ 𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
26 eldifsni 4703 . . . . . . . . . . . . . . . . 17 ((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) → (𝐸𝑗) ≠ ∅)
2726adantr 484 . . . . . . . . . . . . . . . 16 (((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2) → (𝐸𝑗) ≠ ∅)
2827adantr 484 . . . . . . . . . . . . . . 15 ((((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2) ∧ 𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ≠ ∅)
29 eldifsn 4700 . . . . . . . . . . . . . . 15 ((𝐸𝑗) ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ↔ ((𝐸𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}) ∧ (𝐸𝑗) ≠ ∅))
3025, 28, 29sylanbrc 586 . . . . . . . . . . . . . 14 ((((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2) ∧ 𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}))
31 simpr 488 . . . . . . . . . . . . . . 15 (((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2) → (♯‘(𝐸𝑗)) ≤ 2)
3231adantr 484 . . . . . . . . . . . . . 14 ((((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2) ∧ 𝑁 ∉ (𝐸𝑗)) → (♯‘(𝐸𝑗)) ≤ 2)
3312, 30, 32elrabd 3604 . . . . . . . . . . . . 13 ((((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2) ∧ 𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
3433ex 416 . . . . . . . . . . . 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 220 . . . . . . . . . 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 416 . . . . . . . 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 425 . . . . 5 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ((𝑗 ∈ dom 𝐸𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
426, 41syl5bi 245 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → (𝑗𝐹 → (𝐸𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
4342ralrimiv 3104 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ∀𝑗𝐹 (𝐸𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
44 upgruhgr 27193 . . . . . 6 (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
458uhgrfun 27157 . . . . . 6 (𝐺 ∈ UHGraph → Fun 𝐸)
4644, 45syl 17 . . . . 5 (𝐺 ∈ UPGraph → Fun 𝐸)
4746adantr 484 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → Fun 𝐸)
485ssrab3 3995 . . . 4 𝐹 ⊆ dom 𝐸
49 funimass4 6777 . . . 4 ((Fun 𝐸𝐹 ⊆ dom 𝐸) → ((𝐸𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} ↔ ∀𝑗𝐹 (𝐸𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
5047, 48, 49sylancl 589 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ((𝐸𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} ↔ ∀𝑗𝐹 (𝐸𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
5143, 50mpbird 260 . 2 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → (𝐸𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
521, 51eqsstrrid 3950 1 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ran (𝐸𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110  wne 2940  wnel 3046  wral 3061  {crab 3065  cdif 3863  wss 3866  c0 4237  𝒫 cpw 4513  {csn 4541   class class class wbr 5053  dom cdm 5551  ran crn 5552  cres 5553  cima 5554  Fun wfun 6374  wf 6376  cfv 6380  cle 10868  2c2 11885  chash 13896  Vtxcvtx 27087  iEdgciedg 27088  UHGraphcuhgr 27147  UPGraphcupgr 27171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fv 6388  df-uhgr 27149  df-upgr 27173
This theorem is referenced by:  upgrres  27394
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