Theorem upgrreslem 27086

Description: Lemma for upgrres 27088. (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 5568	. 2 ⊢ (𝐸 “ 𝐹) = ran (𝐸 ↾ 𝐹)
2		fveq2 6670	. . . . . . 7 ⊢ (𝑖 = 𝑗 → (𝐸‘𝑖) = (𝐸‘𝑗))
3		neleq2 3129	. . . . . . 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 3683	. . . . 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 26871	. . . . . . 7 ⊢ (𝐺 ∈ UPGraph → 𝐸:dom 𝐸⟶{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
10		ffvelrn 6849	. . . . . . . . . 10 ⊢ ((𝐸:dom 𝐸⟶{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} ∧ 𝑗 ∈ dom 𝐸) → (𝐸‘𝑗) ∈ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
11		fveq2 6670	. . . . . . . . . . . . 13 ⊢ (𝑝 = (𝐸‘𝑗) → (♯‘𝑝) = (♯‘(𝐸‘𝑗)))
12	11	breq1d 5076	. . . . . . . . . . . 12 ⊢ (𝑝 = (𝐸‘𝑗) → ((♯‘𝑝) ≤ 2 ↔ (♯‘(𝐸‘𝑗)) ≤ 2))
13	12	elrab 3680	. . . . . . . . . . 11 ⊢ ((𝐸‘𝑗) ∈ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} ↔ ((𝐸‘𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸‘𝑗)) ≤ 2))
14		eldifsn 4719	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐸‘𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ↔ ((𝐸‘𝑗) ∈ 𝒫 𝑉 ∧ (𝐸‘𝑗) ≠ ∅))
15		simpl 485	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐸‘𝑗) ∈ 𝒫 𝑉 ∧ 𝑁 ∉ (𝐸‘𝑗)) → (𝐸‘𝑗) ∈ 𝒫 𝑉)
16		elpwi 4548	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐸‘𝑗) ∈ 𝒫 𝑉 → (𝐸‘𝑗) ⊆ 𝑉)
17	16	adantr 483	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐸‘𝑗) ∈ 𝒫 𝑉 ∧ 𝑁 ∉ (𝐸‘𝑗)) → (𝐸‘𝑗) ⊆ 𝑉)
18		simpr 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐸‘𝑗) ∈ 𝒫 𝑉 ∧ 𝑁 ∉ (𝐸‘𝑗)) → 𝑁 ∉ (𝐸‘𝑗))
19		elpwdifsn 4721	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐸‘𝑗) ∈ 𝒫 𝑉 ∧ (𝐸‘𝑗) ⊆ 𝑉 ∧ 𝑁 ∉ (𝐸‘𝑗)) → (𝐸‘𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
20	15, 17, 18, 19	syl3anc 1367	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐸‘𝑗) ∈ 𝒫 𝑉 ∧ 𝑁 ∉ (𝐸‘𝑗)) → (𝐸‘𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
21	20	ex 415	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐸‘𝑗) ∈ 𝒫 𝑉 → (𝑁 ∉ (𝐸‘𝑗) → (𝐸‘𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
22	21	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐸‘𝑗) ∈ 𝒫 𝑉 ∧ (𝐸‘𝑗) ≠ ∅) → (𝑁 ∉ (𝐸‘𝑗) → (𝐸‘𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
23	14, 22	sylbi 219	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸‘𝑗) ∈ (𝒫 𝑉 ∖ {∅}) → (𝑁 ∉ (𝐸‘𝑗) → (𝐸‘𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
24	23	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ (((𝐸‘𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸‘𝑗)) ≤ 2) → (𝑁 ∉ (𝐸‘𝑗) → (𝐸‘𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
25	24	imp 409	. . . . . . . . . . . . . . 15 ⊢ ((((𝐸‘𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸‘𝑗)) ≤ 2) ∧ 𝑁 ∉ (𝐸‘𝑗)) → (𝐸‘𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
26		eldifsni 4722	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸‘𝑗) ∈ (𝒫 𝑉 ∖ {∅}) → (𝐸‘𝑗) ≠ ∅)
27	26	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ (((𝐸‘𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸‘𝑗)) ≤ 2) → (𝐸‘𝑗) ≠ ∅)
28	27	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((((𝐸‘𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸‘𝑗)) ≤ 2) ∧ 𝑁 ∉ (𝐸‘𝑗)) → (𝐸‘𝑗) ≠ ∅)
29		eldifsn 4719	. . . . . . . . . . . . . . 15 ⊢ ((𝐸‘𝑗) ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ↔ ((𝐸‘𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}) ∧ (𝐸‘𝑗) ≠ ∅))
30	25, 28, 29	sylanbrc 585	. . . . . . . . . . . . . 14 ⊢ ((((𝐸‘𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸‘𝑗)) ≤ 2) ∧ 𝑁 ∉ (𝐸‘𝑗)) → (𝐸‘𝑗) ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}))
31		simpr 487	. . . . . . . . . . . . . . 15 ⊢ (((𝐸‘𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸‘𝑗)) ≤ 2) → (♯‘(𝐸‘𝑗)) ≤ 2)
32	31	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((((𝐸‘𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸‘𝑗)) ≤ 2) ∧ 𝑁 ∉ (𝐸‘𝑗)) → (♯‘(𝐸‘𝑗)) ≤ 2)
33	12, 30, 32	elrabd 3682	. . . . . . . . . . . . 13 ⊢ ((((𝐸‘𝑗) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸‘𝑗)) ≤ 2) ∧ 𝑁 ∉ (𝐸‘𝑗)) → (𝐸‘𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
34	33	ex 415	. . . . . . . . . . . 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 219	. . . . . . . . . 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 415	. . . . . . . 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 424	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → ((𝑗 ∈ dom 𝐸 ∧ 𝑁 ∉ (𝐸‘𝑗)) → (𝐸‘𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
42	6, 41	syl5bi 244	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → (𝑗 ∈ 𝐹 → (𝐸‘𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
43	42	ralrimiv 3181	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → ∀𝑗 ∈ 𝐹 (𝐸‘𝑗) ∈ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
44		upgruhgr 26887	. . . . . 6 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
45	8	uhgrfun 26851	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸)
46	44, 45	syl 17	. . . . 5 ⊢ (𝐺 ∈ UPGraph → Fun 𝐸)
47	46	adantr 483	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → Fun 𝐸)
48	5	ssrab3 4057	. . . 4 ⊢ 𝐹 ⊆ dom 𝐸
49		funimass4 6730	. . . 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 588	. . 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 259	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → (𝐸 “ 𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
52	1, 51	eqsstrrid 4016	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → ran (𝐸 ↾ 𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ≠ wne 3016   ∉ wnel 3123  ∀wral 3138  {crab 3142   ∖ cdif 3933   ⊆ wss 3936  ∅c0 4291  𝒫 cpw 4539  {csn 4567   class class class wbr 5066  dom cdm 5555  ran crn 5556   ↾ cres 5557   “ cima 5558  Fun wfun 6349  ⟶wf 6351  ‘cfv 6355   ≤ cle 10676  2c2 11693  ♯chash 13691  Vtxcvtx 26781  iEdgciedg 26782  UHGraphcuhgr 26841  UPGraphcupgr 26865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-fv 6363  df-uhgr 26843  df-upgr 26867

This theorem is referenced by:  upgrres  27088