Theorem upgrreslem 26196

Description: Lemma for upgrres 26198. (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 5127	. 2 ⊢ (𝐸 “ 𝐹) = ran (𝐸 ↾ 𝐹)
2		fveq2 6191	. . . . . . 7 ⊢ (𝑖 = 𝑗 → (𝐸‘𝑖) = (𝐸‘𝑗))
3		neleq2 2903	. . . . . . 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 3366	. . . . 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 25981	. . . . . . 7 ⊢ (𝐺 ∈ UPGraph → 𝐸:dom 𝐸⟶{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑝) ≤ 2})
10		ffvelrn 6357	. . . . . . . . . 10 ⊢ ((𝐸:dom 𝐸⟶{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑝) ≤ 2} ∧ 𝑗 ∈ dom 𝐸) → (𝐸‘𝑗) ∈ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑝) ≤ 2})
11		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑝 = (𝐸‘𝑗) → (#‘𝑝) = (#‘(𝐸‘𝑗)))
12	11	breq1d 4663	. . . . . . . . . . . 12 ⊢ (𝑝 = (𝐸‘𝑗) → ((#‘𝑝) ≤ 2 ↔ (#‘(𝐸‘𝑗)) ≤ 2))
13	12	elrab 3363	. . . . . . . . . . 11 ⊢ ((𝐸‘𝑗) ∈ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑝) ≤ 2} ↔ ((𝐸‘𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘(𝐸‘𝑗)) ≤ 2))
14		eldifsn 4317	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐸‘𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ↔ ((𝐸‘𝑗) ∈ 𝒫 𝑉 ∧ (𝐸‘𝑗) ≠ ∅))
15		simpl 473	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐸‘𝑗) ∈ 𝒫 𝑉 ∧ 𝑁 ∉ (𝐸‘𝑗)) → (𝐸‘𝑗) ∈ 𝒫 𝑉)
16		elpwi 4168	. . . . . . . . . . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐸‘𝑗) ∈ 𝒫 𝑉 ∧ 𝑁 ∉ (𝐸‘𝑗)) → 𝑁 ∉ (𝐸‘𝑗))
19		elpwdifsn 4319	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐸‘𝑗) ∈ 𝒫 𝑉 ∧ (𝐸‘𝑗) ⊆ 𝑉 ∧ 𝑁 ∉ (𝐸‘𝑗)) → (𝐸‘𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
20	15, 17, 18, 19	syl3anc 1326	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐸‘𝑗) ∈ 𝒫 𝑉 ∧ 𝑁 ∉ (𝐸‘𝑗)) → (𝐸‘𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
21	20	ex 450	. . . . . . . . . . . . . . . . . . 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 207	. . . . . . . . . . . . . . . . 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 445	. . . . . . . . . . . . . . 15 ⊢ ((((𝐸‘𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘(𝐸‘𝑗)) ≤ 2) ∧ 𝑁 ∉ (𝐸‘𝑗)) → (𝐸‘𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
26		eldifsni 4320	. . . . . . . . . . . . . . . . 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 4317	. . . . . . . . . . . . . . 15 ⊢ ((𝐸‘𝑗) ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ↔ ((𝐸‘𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}) ∧ (𝐸‘𝑗) ≠ ∅))
30	25, 28, 29	sylanbrc 698	. . . . . . . . . . . . . 14 ⊢ ((((𝐸‘𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘(𝐸‘𝑗)) ≤ 2) ∧ 𝑁 ∉ (𝐸‘𝑗)) → (𝐸‘𝑗) ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}))
31		simpr 477	. . . . . . . . . . . . . . 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 3365	. . . . . . . . . . . . 13 ⊢ ((((𝐸‘𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘(𝐸‘𝑗)) ≤ 2) ∧ 𝑁 ∉ (𝐸‘𝑗)) → (𝐸‘𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (#‘𝑝) ≤ 2})
34	33	ex 450	. . . . . . . . . . . 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 207	. . . . . . . . . 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 450	. . . . . . . 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 613	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → ((𝑗 ∈ dom 𝐸 ∧ 𝑁 ∉ (𝐸‘𝑗)) → (𝐸‘𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (#‘𝑝) ≤ 2}))
42	6, 41	syl5bi 232	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → (𝑗 ∈ 𝐹 → (𝐸‘𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (#‘𝑝) ≤ 2}))
43	42	ralrimiv 2965	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → ∀𝑗 ∈ 𝐹 (𝐸‘𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (#‘𝑝) ≤ 2})
44		upgruhgr 25997	. . . . . 6 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph )
45	8	uhgrfun 25961	. . . . . 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		ssrab2 3687	. . . . 5 ⊢ {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)} ⊆ dom 𝐸
49	5, 48	eqsstri 3635	. . . 4 ⊢ 𝐹 ⊆ dom 𝐸
50		funimass4 6247	. . . 4 ⊢ ((Fun 𝐸 ∧ 𝐹 ⊆ dom 𝐸) → ((𝐸 “ 𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (#‘𝑝) ≤ 2} ↔ ∀𝑗 ∈ 𝐹 (𝐸‘𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (#‘𝑝) ≤ 2}))
51	47, 49, 50	sylancl 694	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → ((𝐸 “ 𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (#‘𝑝) ≤ 2} ↔ ∀𝑗 ∈ 𝐹 (𝐸‘𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (#‘𝑝) ≤ 2}))
52	43, 51	mpbird 247	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → (𝐸 “ 𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (#‘𝑝) ≤ 2})
53	1, 52	syl5eqssr 3650	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → ran (𝐸 ↾ 𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (#‘𝑝) ≤ 2})

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794   ∉ wnel 2897  ∀wral 2912  {crab 2916   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  {csn 4177   class class class wbr 4653  dom cdm 5114  ran crn 5115   ↾ cres 5116   “ cima 5117  Fun wfun 5882  ⟶wf 5884  ‘cfv 5888   ≤ cle 10075  2c2 11070  #chash 13117  Vtxcvtx 25874  iEdgciedg 25875   UHGraph cuhgr 25951   UPGraph cupgr 25975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-uhgr 25953  df-upgr 25977

This theorem is referenced by:  upgrres  26198