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Theorem upgrreslem 29336
Description: Lemma for upgrres 29338. (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 5702 . 2 (𝐸𝐹) = ran (𝐸𝐹)
2 fveq2 6907 . . . . . . 7 (𝑖 = 𝑗 → (𝐸𝑖) = (𝐸𝑗))
3 neleq2 3051 . . . . . . 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 3698 . . . . 5 (𝑗𝐹 ↔ (𝑗 ∈ dom 𝐸𝑁 ∉ (𝐸𝑗)))
7 upgrres.v . . . . . . . 8 𝑉 = (Vtx‘𝐺)
8 upgrres.e . . . . . . . 8 𝐸 = (iEdg‘𝐺)
97, 8upgrf 29118 . . . . . . 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 6907 . . . . . . . . . . . . 13 (𝑝 = (𝐸𝑗) → (♯‘𝑝) = (♯‘(𝐸𝑗)))
1211breq1d 5158 . . . . . . . . . . . 12 (𝑝 = (𝐸𝑗) → ((♯‘𝑝) ≤ 2 ↔ (♯‘(𝐸𝑗)) ≤ 2))
1312elrab 3695 . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . 21 (((𝐸𝑗) ∈ 𝒫 𝑉𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ∈ 𝒫 𝑉)
16 elpwi 4612 . . . . . . . . . . . . . . . . . . . . . 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 4794 . . . . . . . . . . . . . . . . . . . . 21 (((𝐸𝑗) ∈ 𝒫 𝑉 ∧ (𝐸𝑗) ⊆ 𝑉𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
2015, 17, 18, 19syl3anc 1370 . . . . . . . . . . . . . . . . . . . 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 4795 . . . . . . . . . . . . . . . . 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 4791 . . . . . . . . . . . . . . 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 3697 . . . . . . . . . . . . 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 3143 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ∀𝑗𝐹 (𝐸𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
44 upgruhgr 29134 . . . . . 6 (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
458uhgrfun 29098 . . . . . 6 (𝐺 ∈ UHGraph → Fun 𝐸)
4644, 45syl 17 . . . . 5 (𝐺 ∈ UPGraph → Fun 𝐸)
4746adantr 480 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → Fun 𝐸)
485ssrab3 4092 . . . 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 4045 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 1537  wcel 2106  wne 2938  wnel 3044  wral 3059  {crab 3433  cdif 3960  wss 3963  c0 4339  𝒫 cpw 4605  {csn 4631   class class class wbr 5148  dom cdm 5689  ran crn 5690  cres 5691  cima 5692  Fun wfun 6557  wf 6559  cfv 6563  cle 11294  2c2 12319  chash 14366  Vtxcvtx 29028  iEdgciedg 29029  UHGraphcuhgr 29088  UPGraphcupgr 29112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-uhgr 29090  df-upgr 29114
This theorem is referenced by:  upgrres  29338
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