Theorem upgrreslem 28550

Description: Lemma for upgrres 28552. (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 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}

Assertion

Ref	Expression
upgrreslem	⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → ran (𝐸 ↾ 𝐹) ⊆ {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.

Step	Hyp	Ref	Expression
1		df-ima 5688	. 2 ⊢ (𝐸 “ 𝐹) = ran (𝐸 ↾ 𝐹)
2		fveq2 6888	. . . . . . 7 ⊢ (𝑖 = 𝑗 → (𝐸‘𝑖) = (𝐸‘𝑗))
3		neleq2 3053	. . . . . . 7 ⊢ ((𝐸‘𝑖) = (𝐸‘𝑗) → (𝑁 ∉ (𝐸‘𝑖) ↔ 𝑁 ∉ (𝐸‘𝑗)))
4	2, 3	syl 17	. . . . . 6 ⊢ (𝑖 = 𝑗 → (𝑁 ∉ (𝐸‘𝑖) ↔ 𝑁 ∉ (𝐸‘𝑗)))
5		upgrres.f	. . . . . 6 ⊢ 𝐹 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}
6	4, 5	elrab2 3685	. . . . 5 ⊢ (𝑗 ∈ 𝐹 ↔ (𝑗 ∈ dom 𝐸 ∧ 𝑁 ∉ (𝐸‘𝑗)))
7		upgrres.v	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
8		upgrres.e	. . . . . . . 8 ⊢ 𝐸 = (iEdg‘𝐺)
9	7, 8	upgrf 28335	. . . . . . 7 ⊢ (𝐺 ∈ UPGraph → 𝐸:dom 𝐸⟶{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
10		ffvelcdm 7080	. . . . . . . . . 10 ⊢ ((𝐸:dom 𝐸⟶{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} ∧ 𝑗 ∈ dom 𝐸) → (𝐸‘𝑗) ∈ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
11		fveq2 6888	. . . . . . . . . . . . 13 ⊢ (𝑝 = (𝐸‘𝑗) → (♯‘𝑝) = (♯‘(𝐸‘𝑗)))
12	11	breq1d 5157	. . . . . . . . . . . 12 ⊢ (𝑝 = (𝐸‘𝑗) → ((♯‘𝑝) ≤ 2 ↔ (♯‘(𝐸‘𝑗)) ≤ 2))
13	12	elrab 3682	. . . . . . . . . . 11 ⊢ ((𝐸‘𝑗) ∈ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} ↔ ((𝐸‘𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸‘𝑗)) ≤ 2))
14		eldifsn 4789	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐸‘𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ↔ ((𝐸‘𝑗) ∈ 𝒫 𝑉 ∧ (𝐸‘𝑗) ≠ ∅))
15		simpl 483	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐸‘𝑗) ∈ 𝒫 𝑉 ∧ 𝑁 ∉ (𝐸‘𝑗)) → (𝐸‘𝑗) ∈ 𝒫 𝑉)
16		elpwi 4608	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐸‘𝑗) ∈ 𝒫 𝑉 → (𝐸‘𝑗) ⊆ 𝑉)
17	16	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐸‘𝑗) ∈ 𝒫 𝑉 ∧ 𝑁 ∉ (𝐸‘𝑗)) → (𝐸‘𝑗) ⊆ 𝑉)
18		simpr 485	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐸‘𝑗) ∈ 𝒫 𝑉 ∧ 𝑁 ∉ (𝐸‘𝑗)) → 𝑁 ∉ (𝐸‘𝑗))
19		elpwdifsn 4791	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐸‘𝑗) ∈ 𝒫 𝑉 ∧ (𝐸‘𝑗) ⊆ 𝑉 ∧ 𝑁 ∉ (𝐸‘𝑗)) → (𝐸‘𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
20	15, 17, 18, 19	syl3anc 1371	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐸‘𝑗) ∈ 𝒫 𝑉 ∧ 𝑁 ∉ (𝐸‘𝑗)) → (𝐸‘𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
21	20	ex 413	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐸‘𝑗) ∈ 𝒫 𝑉 → (𝑁 ∉ (𝐸‘𝑗) → (𝐸‘𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
22	21	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐸‘𝑗) ∈ 𝒫 𝑉 ∧ (𝐸‘𝑗) ≠ ∅) → (𝑁 ∉ (𝐸‘𝑗) → (𝐸‘𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
23	14, 22	sylbi 216	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸‘𝑗) ∈ (𝒫 𝑉 ∖ {∅}) → (𝑁 ∉ (𝐸‘𝑗) → (𝐸‘𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
24	23	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝐸‘𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸‘𝑗)) ≤ 2) → (𝑁 ∉ (𝐸‘𝑗) → (𝐸‘𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
25	24	imp 407	. . . . . . . . . . . . . . 15 ⊢ ((((𝐸‘𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸‘𝑗)) ≤ 2) ∧ 𝑁 ∉ (𝐸‘𝑗)) → (𝐸‘𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
26		eldifsni 4792	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸‘𝑗) ∈ (𝒫 𝑉 ∖ {∅}) → (𝐸‘𝑗) ≠ ∅)
27	26	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝐸‘𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸‘𝑗)) ≤ 2) → (𝐸‘𝑗) ≠ ∅)
28	27	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝐸‘𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸‘𝑗)) ≤ 2) ∧ 𝑁 ∉ (𝐸‘𝑗)) → (𝐸‘𝑗) ≠ ∅)
29		eldifsn 4789	. . . . . . . . . . . . . . 15 ⊢ ((𝐸‘𝑗) ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ↔ ((𝐸‘𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}) ∧ (𝐸‘𝑗) ≠ ∅))
30	25, 28, 29	sylanbrc 583	. . . . . . . . . . . . . 14 ⊢ ((((𝐸‘𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸‘𝑗)) ≤ 2) ∧ 𝑁 ∉ (𝐸‘𝑗)) → (𝐸‘𝑗) ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}))
31		simpr 485	. . . . . . . . . . . . . . 15 ⊢ (((𝐸‘𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸‘𝑗)) ≤ 2) → (♯‘(𝐸‘𝑗)) ≤ 2)
32	31	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝐸‘𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸‘𝑗)) ≤ 2) ∧ 𝑁 ∉ (𝐸‘𝑗)) → (♯‘(𝐸‘𝑗)) ≤ 2)
33	12, 30, 32	elrabd 3684	. . . . . . . . . . . . 13 ⊢ ((((𝐸‘𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸‘𝑗)) ≤ 2) ∧ 𝑁 ∉ (𝐸‘𝑗)) → (𝐸‘𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
34	33	ex 413	. . . . . . . . . . . 12 ⊢ (((𝐸‘𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸‘𝑗)) ≤ 2) → (𝑁 ∉ (𝐸‘𝑗) → (𝐸‘𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
35	34	a1d 25	. . . . . . . . . . 11 ⊢ (((𝐸‘𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸‘𝑗)) ≤ 2) → (𝑁 ∈ 𝑉 → (𝑁 ∉ (𝐸‘𝑗) → (𝐸‘𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})))
36	13, 35	sylbi 216	. . . . . . . . . 10 ⊢ ((𝐸‘𝑗) ∈ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} → (𝑁 ∈ 𝑉 → (𝑁 ∉ (𝐸‘𝑗) → (𝐸‘𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})))
37	10, 36	syl 17	. . . . . . . . 9 ⊢ ((𝐸:dom 𝐸⟶{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} ∧ 𝑗 ∈ dom 𝐸) → (𝑁 ∈ 𝑉 → (𝑁 ∉ (𝐸‘𝑗) → (𝐸‘𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})))
38	37	ex 413	. . . . . . . 8 ⊢ (𝐸:dom 𝐸⟶{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} → (𝑗 ∈ dom 𝐸 → (𝑁 ∈ 𝑉 → (𝑁 ∉ (𝐸‘𝑗) → (𝐸‘𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))))
39	38	com23 86	. . . . . . 7 ⊢ (𝐸:dom 𝐸⟶{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} → (𝑁 ∈ 𝑉 → (𝑗 ∈ dom 𝐸 → (𝑁 ∉ (𝐸‘𝑗) → (𝐸‘𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))))
40	9, 39	syl 17	. . . . . 6 ⊢ (𝐺 ∈ UPGraph → (𝑁 ∈ 𝑉 → (𝑗 ∈ dom 𝐸 → (𝑁 ∉ (𝐸‘𝑗) → (𝐸‘𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))))
41	40	imp4b 422	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → ((𝑗 ∈ dom 𝐸 ∧ 𝑁 ∉ (𝐸‘𝑗)) → (𝐸‘𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
42	6, 41	biimtrid 241	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → (𝑗 ∈ 𝐹 → (𝐸‘𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
43	42	ralrimiv 3145	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → ∀𝑗 ∈ 𝐹 (𝐸‘𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
44		upgruhgr 28351	. . . . . 6 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
45	8	uhgrfun 28315	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸)
46	44, 45	syl 17	. . . . 5 ⊢ (𝐺 ∈ UPGraph → Fun 𝐸)
47	46	adantr 481	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → Fun 𝐸)
48	5	ssrab3 4079	. . . 4 ⊢ 𝐹 ⊆ dom 𝐸
49		funimass4 6953	. . . 4 ⊢ ((Fun 𝐸 ∧ 𝐹 ⊆ dom 𝐸) → ((𝐸 “ 𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} ↔ ∀𝑗 ∈ 𝐹 (𝐸‘𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
50	47, 48, 49	sylancl 586	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → ((𝐸 “ 𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} ↔ ∀𝑗 ∈ 𝐹 (𝐸‘𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
51	43, 50	mpbird 256	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → (𝐸 “ 𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
52	1, 51	eqsstrrid 4030	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → ran (𝐸 ↾ 𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2940   ∉ wnel 3046  ∀wral 3061  {crab 3432   ∖ cdif 3944   ⊆ wss 3947  ∅c0 4321  𝒫 cpw 4601  {csn 4627   class class class wbr 5147  dom cdm 5675  ran crn 5676   ↾ cres 5677   “ cima 5678  Fun wfun 6534  ⟶wf 6536  ‘cfv 6540   ≤ cle 11245  2c2 12263  ♯chash 14286  Vtxcvtx 28245  iEdgciedg 28246  UHGraphcuhgr 28305  UPGraphcupgr 28329

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-uhgr 28307  df-upgr 28331

This theorem is referenced by:  upgrres  28552