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Theorem umgrreslem 26242
Description: Lemma for umgrres 26244 and usgrres 26245. (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
umgrreslem ((𝐺 ∈ UMGraph ∧ 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 umgrreslem
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 df-ima 5156 . 2 (𝐸𝐹) = ran (𝐸𝐹)
2 fveq2 6229 . . . . . . 7 (𝑖 = 𝑗 → (𝐸𝑖) = (𝐸𝑗))
3 neleq2 2932 . . . . . . 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 3399 . . . . 5 (𝑗𝐹 ↔ (𝑗 ∈ dom 𝐸𝑁 ∉ (𝐸𝑗)))
7 upgrres.v . . . . . . . 8 𝑉 = (Vtx‘𝐺)
8 upgrres.e . . . . . . . 8 𝐸 = (iEdg‘𝐺)
97, 8umgrf 26038 . . . . . . 7 (𝐺 ∈ UMGraph → 𝐸:dom 𝐸⟶{𝑝 ∈ 𝒫 𝑉 ∣ (#‘𝑝) = 2})
10 ffvelrn 6397 . . . . . . . . . 10 ((𝐸:dom 𝐸⟶{𝑝 ∈ 𝒫 𝑉 ∣ (#‘𝑝) = 2} ∧ 𝑗 ∈ dom 𝐸) → (𝐸𝑗) ∈ {𝑝 ∈ 𝒫 𝑉 ∣ (#‘𝑝) = 2})
11 fveq2 6229 . . . . . . . . . . . . 13 (𝑝 = (𝐸𝑗) → (#‘𝑝) = (#‘(𝐸𝑗)))
1211eqeq1d 2653 . . . . . . . . . . . 12 (𝑝 = (𝐸𝑗) → ((#‘𝑝) = 2 ↔ (#‘(𝐸𝑗)) = 2))
1312elrab 3396 . . . . . . . . . . 11 ((𝐸𝑗) ∈ {𝑝 ∈ 𝒫 𝑉 ∣ (#‘𝑝) = 2} ↔ ((𝐸𝑗) ∈ 𝒫 𝑉 ∧ (#‘(𝐸𝑗)) = 2))
14 simpll 805 . . . . . . . . . . . . . . 15 ((((𝐸𝑗) ∈ 𝒫 𝑉 ∧ (#‘(𝐸𝑗)) = 2) ∧ 𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ∈ 𝒫 𝑉)
15 elpwi 4201 . . . . . . . . . . . . . . . . 17 ((𝐸𝑗) ∈ 𝒫 𝑉 → (𝐸𝑗) ⊆ 𝑉)
1615adantr 480 . . . . . . . . . . . . . . . 16 (((𝐸𝑗) ∈ 𝒫 𝑉 ∧ (#‘(𝐸𝑗)) = 2) → (𝐸𝑗) ⊆ 𝑉)
1716adantr 480 . . . . . . . . . . . . . . 15 ((((𝐸𝑗) ∈ 𝒫 𝑉 ∧ (#‘(𝐸𝑗)) = 2) ∧ 𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ⊆ 𝑉)
18 simpr 476 . . . . . . . . . . . . . . 15 ((((𝐸𝑗) ∈ 𝒫 𝑉 ∧ (#‘(𝐸𝑗)) = 2) ∧ 𝑁 ∉ (𝐸𝑗)) → 𝑁 ∉ (𝐸𝑗))
19 elpwdifsn 4352 . . . . . . . . . . . . . . 15 (((𝐸𝑗) ∈ 𝒫 𝑉 ∧ (𝐸𝑗) ⊆ 𝑉𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
2014, 17, 18, 19syl3anc 1366 . . . . . . . . . . . . . 14 ((((𝐸𝑗) ∈ 𝒫 𝑉 ∧ (#‘(𝐸𝑗)) = 2) ∧ 𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
21 simpr 476 . . . . . . . . . . . . . . 15 (((𝐸𝑗) ∈ 𝒫 𝑉 ∧ (#‘(𝐸𝑗)) = 2) → (#‘(𝐸𝑗)) = 2)
2221adantr 480 . . . . . . . . . . . . . 14 ((((𝐸𝑗) ∈ 𝒫 𝑉 ∧ (#‘(𝐸𝑗)) = 2) ∧ 𝑁 ∉ (𝐸𝑗)) → (#‘(𝐸𝑗)) = 2)
2312, 20, 22elrabd 3398 . . . . . . . . . . . . 13 ((((𝐸𝑗) ∈ 𝒫 𝑉 ∧ (#‘(𝐸𝑗)) = 2) ∧ 𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ∈ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑝) = 2})
2423ex 449 . . . . . . . . . . . 12 (((𝐸𝑗) ∈ 𝒫 𝑉 ∧ (#‘(𝐸𝑗)) = 2) → (𝑁 ∉ (𝐸𝑗) → (𝐸𝑗) ∈ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑝) = 2}))
2524a1d 25 . . . . . . . . . . 11 (((𝐸𝑗) ∈ 𝒫 𝑉 ∧ (#‘(𝐸𝑗)) = 2) → (𝑁𝑉 → (𝑁 ∉ (𝐸𝑗) → (𝐸𝑗) ∈ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑝) = 2})))
2613, 25sylbi 207 . . . . . . . . . 10 ((𝐸𝑗) ∈ {𝑝 ∈ 𝒫 𝑉 ∣ (#‘𝑝) = 2} → (𝑁𝑉 → (𝑁 ∉ (𝐸𝑗) → (𝐸𝑗) ∈ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑝) = 2})))
2710, 26syl 17 . . . . . . . . 9 ((𝐸:dom 𝐸⟶{𝑝 ∈ 𝒫 𝑉 ∣ (#‘𝑝) = 2} ∧ 𝑗 ∈ dom 𝐸) → (𝑁𝑉 → (𝑁 ∉ (𝐸𝑗) → (𝐸𝑗) ∈ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑝) = 2})))
2827ex 449 . . . . . . . 8 (𝐸:dom 𝐸⟶{𝑝 ∈ 𝒫 𝑉 ∣ (#‘𝑝) = 2} → (𝑗 ∈ dom 𝐸 → (𝑁𝑉 → (𝑁 ∉ (𝐸𝑗) → (𝐸𝑗) ∈ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑝) = 2}))))
2928com23 86 . . . . . . 7 (𝐸:dom 𝐸⟶{𝑝 ∈ 𝒫 𝑉 ∣ (#‘𝑝) = 2} → (𝑁𝑉 → (𝑗 ∈ dom 𝐸 → (𝑁 ∉ (𝐸𝑗) → (𝐸𝑗) ∈ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑝) = 2}))))
309, 29syl 17 . . . . . 6 (𝐺 ∈ UMGraph → (𝑁𝑉 → (𝑗 ∈ dom 𝐸 → (𝑁 ∉ (𝐸𝑗) → (𝐸𝑗) ∈ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑝) = 2}))))
3130imp4b 612 . . . . 5 ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → ((𝑗 ∈ dom 𝐸𝑁 ∉ (𝐸𝑗)) → (𝐸𝑗) ∈ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑝) = 2}))
326, 31syl5bi 232 . . . 4 ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → (𝑗𝐹 → (𝐸𝑗) ∈ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑝) = 2}))
3332ralrimiv 2994 . . 3 ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → ∀𝑗𝐹 (𝐸𝑗) ∈ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑝) = 2})
34 umgruhgr 26044 . . . . . 6 (𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph)
358uhgrfun 26006 . . . . . 6 (𝐺 ∈ UHGraph → Fun 𝐸)
3634, 35syl 17 . . . . 5 (𝐺 ∈ UMGraph → Fun 𝐸)
3736adantr 480 . . . 4 ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → Fun 𝐸)
38 ssrab2 3720 . . . . 5 {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)} ⊆ dom 𝐸
395, 38eqsstri 3668 . . . 4 𝐹 ⊆ dom 𝐸
40 funimass4 6286 . . . 4 ((Fun 𝐸𝐹 ⊆ dom 𝐸) → ((𝐸𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑝) = 2} ↔ ∀𝑗𝐹 (𝐸𝑗) ∈ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑝) = 2}))
4137, 39, 40sylancl 695 . . 3 ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → ((𝐸𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑝) = 2} ↔ ∀𝑗𝐹 (𝐸𝑗) ∈ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑝) = 2}))
4233, 41mpbird 247 . 2 ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → (𝐸𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑝) = 2})
431, 42syl5eqssr 3683 1 ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → ran (𝐸𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑝) = 2})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wnel 2926  wral 2941  {crab 2945  cdif 3604  wss 3607  𝒫 cpw 4191  {csn 4210  dom cdm 5143  ran crn 5144  cres 5145  cima 5146  Fun wfun 5920  wf 5922  cfv 5926  2c2 11108  #chash 13157  Vtxcvtx 25919  iEdgciedg 25920  UHGraphcuhgr 25996  UMGraphcumgr 26021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-hash 13158  df-uhgr 25998  df-upgr 26022  df-umgr 26023
This theorem is referenced by:  umgrres  26244  usgrres  26245
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