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Theorem upgrreslem 27013
Description: Lemma for upgrres 27015. (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 5561 . 2 (𝐸𝐹) = ran (𝐸𝐹)
2 fveq2 6663 . . . . . . 7 (𝑖 = 𝑗 → (𝐸𝑖) = (𝐸𝑗))
3 neleq2 3126 . . . . . . 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 3680 . . . . 5 (𝑗𝐹 ↔ (𝑗 ∈ dom 𝐸𝑁 ∉ (𝐸𝑗)))
7 upgrres.v . . . . . . . 8 𝑉 = (Vtx‘𝐺)
8 upgrres.e . . . . . . . 8 𝐸 = (iEdg‘𝐺)
97, 8upgrf 26798 . . . . . . 7 (𝐺 ∈ UPGraph → 𝐸:dom 𝐸⟶{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
10 ffvelrn 6841 . . . . . . . . . 10 ((𝐸:dom 𝐸⟶{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} ∧ 𝑗 ∈ dom 𝐸) → (𝐸𝑗) ∈ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
11 fveq2 6663 . . . . . . . . . . . . 13 (𝑝 = (𝐸𝑗) → (♯‘𝑝) = (♯‘(𝐸𝑗)))
1211breq1d 5067 . . . . . . . . . . . 12 (𝑝 = (𝐸𝑗) → ((♯‘𝑝) ≤ 2 ↔ (♯‘(𝐸𝑗)) ≤ 2))
1312elrab 3677 . . . . . . . . . . 11 ((𝐸𝑗) ∈ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} ↔ ((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2))
14 eldifsn 4711 . . . . . . . . . . . . . . . . . 18 ((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ↔ ((𝐸𝑗) ∈ 𝒫 𝑉 ∧ (𝐸𝑗) ≠ ∅))
15 simpl 483 . . . . . . . . . . . . . . . . . . . . 21 (((𝐸𝑗) ∈ 𝒫 𝑉𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ∈ 𝒫 𝑉)
16 elpwi 4547 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐸𝑗) ∈ 𝒫 𝑉 → (𝐸𝑗) ⊆ 𝑉)
1716adantr 481 . . . . . . . . . . . . . . . . . . . . 21 (((𝐸𝑗) ∈ 𝒫 𝑉𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ⊆ 𝑉)
18 simpr 485 . . . . . . . . . . . . . . . . . . . . 21 (((𝐸𝑗) ∈ 𝒫 𝑉𝑁 ∉ (𝐸𝑗)) → 𝑁 ∉ (𝐸𝑗))
19 elpwdifsn 4713 . . . . . . . . . . . . . . . . . . . . 21 (((𝐸𝑗) ∈ 𝒫 𝑉 ∧ (𝐸𝑗) ⊆ 𝑉𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
2015, 17, 18, 19syl3anc 1363 . . . . . . . . . . . . . . . . . . . 20 (((𝐸𝑗) ∈ 𝒫 𝑉𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
2120ex 413 . . . . . . . . . . . . . . . . . . 19 ((𝐸𝑗) ∈ 𝒫 𝑉 → (𝑁 ∉ (𝐸𝑗) → (𝐸𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
2221adantr 481 . . . . . . . . . . . . . . . . . 18 (((𝐸𝑗) ∈ 𝒫 𝑉 ∧ (𝐸𝑗) ≠ ∅) → (𝑁 ∉ (𝐸𝑗) → (𝐸𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
2314, 22sylbi 218 . . . . . . . . . . . . . . . . 17 ((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) → (𝑁 ∉ (𝐸𝑗) → (𝐸𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
2423adantr 481 . . . . . . . . . . . . . . . 16 (((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2) → (𝑁 ∉ (𝐸𝑗) → (𝐸𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
2524imp 407 . . . . . . . . . . . . . . 15 ((((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2) ∧ 𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
26 eldifsni 4714 . . . . . . . . . . . . . . . . 17 ((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) → (𝐸𝑗) ≠ ∅)
2726adantr 481 . . . . . . . . . . . . . . . 16 (((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2) → (𝐸𝑗) ≠ ∅)
2827adantr 481 . . . . . . . . . . . . . . 15 ((((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2) ∧ 𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ≠ ∅)
29 eldifsn 4711 . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . 15 (((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2) → (♯‘(𝐸𝑗)) ≤ 2)
3231adantr 481 . . . . . . . . . . . . . 14 ((((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2) ∧ 𝑁 ∉ (𝐸𝑗)) → (♯‘(𝐸𝑗)) ≤ 2)
3312, 30, 32elrabd 3679 . . . . . . . . . . . . 13 ((((𝐸𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝑗)) ≤ 2) ∧ 𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
3433ex 413 . . . . . . . . . . . 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 218 . . . . . . . . . 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 413 . . . . . . . 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 422 . . . . 5 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ((𝑗 ∈ dom 𝐸𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
426, 41syl5bi 243 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → (𝑗𝐹 → (𝐸𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
4342ralrimiv 3178 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ∀𝑗𝐹 (𝐸𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
44 upgruhgr 26814 . . . . . 6 (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
458uhgrfun 26778 . . . . . 6 (𝐺 ∈ UHGraph → Fun 𝐸)
4644, 45syl 17 . . . . 5 (𝐺 ∈ UPGraph → Fun 𝐸)
4746adantr 481 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → Fun 𝐸)
485ssrab3 4054 . . . 4 𝐹 ⊆ dom 𝐸
49 funimass4 6723 . . . 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 258 . 2 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → (𝐸𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
521, 51eqsstrrid 4013 1 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ran (𝐸𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wne 3013  wnel 3120  wral 3135  {crab 3139  cdif 3930  wss 3933  c0 4288  𝒫 cpw 4535  {csn 4557   class class class wbr 5057  dom cdm 5548  ran crn 5549  cres 5550  cima 5551  Fun wfun 6342  wf 6344  cfv 6348  cle 10664  2c2 11680  chash 13678  Vtxcvtx 26708  iEdgciedg 26709  UHGraphcuhgr 26768  UPGraphcupgr 26792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-uhgr 26770  df-upgr 26794
This theorem is referenced by:  upgrres  27015
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