Theorem cusgrfilem2 28980

Description: Lemma 2 for cusgrfi 28982. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.)

Hypotheses

Ref	Expression
cusgrfi.v	⊢ 𝑉 = (Vtx‘𝐺)
cusgrfi.p	⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁})}
cusgrfi.f	⊢ 𝐹 = (𝑥 ∈ (𝑉 ∖ {𝑁}) ↦ {𝑥, 𝑁})

Assertion

Ref	Expression
cusgrfilem2	⊢ (𝑁 ∈ 𝑉 → 𝐹:(𝑉 ∖ {𝑁})–1-1-onto→𝑃)

Distinct variable groups:   𝑥,𝐺   𝑁,𝑎,𝑥   𝑉,𝑎,𝑥   𝑥,𝑃

Allowed substitution hints:   𝑃(𝑎)   𝐹(𝑥,𝑎)   𝐺(𝑎)

Proof of Theorem cusgrfilem2

Dummy variables 𝑒 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldifi 4125	. . . . 5 ⊢ (𝑥 ∈ (𝑉 ∖ {𝑁}) → 𝑥 ∈ 𝑉)
2		id 22	. . . . 5 ⊢ (𝑁 ∈ 𝑉 → 𝑁 ∈ 𝑉)
3		prelpwi 5446	. . . . 5 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) → {𝑥, 𝑁} ∈ 𝒫 𝑉)
4	1, 2, 3	syl2anr 595	. . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → {𝑥, 𝑁} ∈ 𝒫 𝑉)
5	1	adantl 480	. . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → 𝑥 ∈ 𝑉)
6		eldifsni 4792	. . . . . . 7 ⊢ (𝑥 ∈ (𝑉 ∖ {𝑁}) → 𝑥 ≠ 𝑁)
7	6	adantl 480	. . . . . 6 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → 𝑥 ≠ 𝑁)
8		eqidd 2731	. . . . . 6 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → {𝑥, 𝑁} = {𝑥, 𝑁})
9	7, 8	jca 510	. . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑥 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑥, 𝑁}))
10		id 22	. . . . . 6 ⊢ (𝑥 ∈ 𝑉 → 𝑥 ∈ 𝑉)
11		neeq1 3001	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → (𝑎 ≠ 𝑁 ↔ 𝑥 ≠ 𝑁))
12		preq1 4736	. . . . . . . . 9 ⊢ (𝑎 = 𝑥 → {𝑎, 𝑁} = {𝑥, 𝑁})
13	12	eqeq2d 2741	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → ({𝑥, 𝑁} = {𝑎, 𝑁} ↔ {𝑥, 𝑁} = {𝑥, 𝑁}))
14	11, 13	anbi12d 629	. . . . . . 7 ⊢ (𝑎 = 𝑥 → ((𝑎 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁}) ↔ (𝑥 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑥, 𝑁})))
15	14	adantl 480	. . . . . 6 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑎 = 𝑥) → ((𝑎 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁}) ↔ (𝑥 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑥, 𝑁})))
16	10, 15	rspcedv 3604	. . . . 5 ⊢ (𝑥 ∈ 𝑉 → ((𝑥 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑥, 𝑁}) → ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁})))
17	5, 9, 16	sylc 65	. . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁}))
18		cusgrfi.p	. . . . . 6 ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁})}
19	18	eleq2i 2823	. . . . 5 ⊢ ({𝑥, 𝑁} ∈ 𝑃 ↔ {𝑥, 𝑁} ∈ {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁})})
20		eqeq1 2734	. . . . . . . 8 ⊢ (𝑣 = {𝑥, 𝑁} → (𝑣 = {𝑎, 𝑁} ↔ {𝑥, 𝑁} = {𝑎, 𝑁}))
21	20	anbi2d 627	. . . . . . 7 ⊢ (𝑣 = {𝑥, 𝑁} → ((𝑎 ≠ 𝑁 ∧ 𝑣 = {𝑎, 𝑁}) ↔ (𝑎 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁})))
22	21	rexbidv 3176	. . . . . 6 ⊢ (𝑣 = {𝑥, 𝑁} → (∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑣 = {𝑎, 𝑁}) ↔ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁})))
23		eqeq1 2734	. . . . . . . . 9 ⊢ (𝑥 = 𝑣 → (𝑥 = {𝑎, 𝑁} ↔ 𝑣 = {𝑎, 𝑁}))
24	23	anbi2d 627	. . . . . . . 8 ⊢ (𝑥 = 𝑣 → ((𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}) ↔ (𝑎 ≠ 𝑁 ∧ 𝑣 = {𝑎, 𝑁})))
25	24	rexbidv 3176	. . . . . . 7 ⊢ (𝑥 = 𝑣 → (∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}) ↔ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑣 = {𝑎, 𝑁})))
26	25	cbvrabv 3440	. . . . . 6 ⊢ {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁})} = {𝑣 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑣 = {𝑎, 𝑁})}
27	22, 26	elrab2 3685	. . . . 5 ⊢ ({𝑥, 𝑁} ∈ {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁})} ↔ ({𝑥, 𝑁} ∈ 𝒫 𝑉 ∧ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁})))
28	19, 27	bitri 274	. . . 4 ⊢ ({𝑥, 𝑁} ∈ 𝑃 ↔ ({𝑥, 𝑁} ∈ 𝒫 𝑉 ∧ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁})))
29	4, 17, 28	sylanbrc 581	. . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → {𝑥, 𝑁} ∈ 𝑃)
30	29	ralrimiva 3144	. 2 ⊢ (𝑁 ∈ 𝑉 → ∀𝑥 ∈ (𝑉 ∖ {𝑁}){𝑥, 𝑁} ∈ 𝑃)
31		simpl 481	. . . . . . . . . . 11 ⊢ ((𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}) → 𝑎 ≠ 𝑁)
32	31	anim2i 615	. . . . . . . . . 10 ⊢ ((𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁})) → (𝑎 ∈ 𝑉 ∧ 𝑎 ≠ 𝑁))
33	32	adantl 480	. . . . . . . . 9 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝑒 ∈ 𝒫 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}))) → (𝑎 ∈ 𝑉 ∧ 𝑎 ≠ 𝑁))
34		eldifsn 4789	. . . . . . . . 9 ⊢ (𝑎 ∈ (𝑉 ∖ {𝑁}) ↔ (𝑎 ∈ 𝑉 ∧ 𝑎 ≠ 𝑁))
35	33, 34	sylibr 233	. . . . . . . 8 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝑒 ∈ 𝒫 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}))) → 𝑎 ∈ (𝑉 ∖ {𝑁}))
36		eqeq1 2734	. . . . . . . . . . . . . 14 ⊢ (𝑒 = {𝑎, 𝑁} → (𝑒 = {𝑥, 𝑁} ↔ {𝑎, 𝑁} = {𝑥, 𝑁}))
37	36	adantl 480	. . . . . . . . . . . . 13 ⊢ ((𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}) → (𝑒 = {𝑥, 𝑁} ↔ {𝑎, 𝑁} = {𝑥, 𝑁}))
38	37	ad2antlr 723	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁})) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑒 = {𝑥, 𝑁} ↔ {𝑎, 𝑁} = {𝑥, 𝑁}))
39		vex 3476	. . . . . . . . . . . . . 14 ⊢ 𝑎 ∈ V
40		vex 3476	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
41	39, 40	preqr1 4848	. . . . . . . . . . . . 13 ⊢ ({𝑎, 𝑁} = {𝑥, 𝑁} → 𝑎 = 𝑥)
42	41	equcomd 2020	. . . . . . . . . . . 12 ⊢ ({𝑎, 𝑁} = {𝑥, 𝑁} → 𝑥 = 𝑎)
43	38, 42	syl6bi 252	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁})) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑒 = {𝑥, 𝑁} → 𝑥 = 𝑎))
44	43	adantll 710	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ 𝑉 ∧ 𝑒 ∈ 𝒫 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}))) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑒 = {𝑥, 𝑁} → 𝑥 = 𝑎))
45	12	equcoms 2021	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑎 → {𝑎, 𝑁} = {𝑥, 𝑁})
46	45	eqeq2d 2741	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑎 → (𝑒 = {𝑎, 𝑁} ↔ 𝑒 = {𝑥, 𝑁}))
47	46	biimpcd 248	. . . . . . . . . . . . 13 ⊢ (𝑒 = {𝑎, 𝑁} → (𝑥 = 𝑎 → 𝑒 = {𝑥, 𝑁}))
48	47	adantl 480	. . . . . . . . . . . 12 ⊢ ((𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}) → (𝑥 = 𝑎 → 𝑒 = {𝑥, 𝑁}))
49	48	adantl 480	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁})) → (𝑥 = 𝑎 → 𝑒 = {𝑥, 𝑁}))
50	49	ad2antlr 723	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ 𝑉 ∧ 𝑒 ∈ 𝒫 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}))) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑥 = 𝑎 → 𝑒 = {𝑥, 𝑁}))
51	44, 50	impbid 211	. . . . . . . . 9 ⊢ ((((𝑁 ∈ 𝑉 ∧ 𝑒 ∈ 𝒫 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}))) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎))
52	51	ralrimiva 3144	. . . . . . . 8 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝑒 ∈ 𝒫 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}))) → ∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎))
53	35, 52	jca 510	. . . . . . 7 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝑒 ∈ 𝒫 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}))) → (𝑎 ∈ (𝑉 ∖ {𝑁}) ∧ ∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎)))
54	53	ex 411	. . . . . 6 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑒 ∈ 𝒫 𝑉) → ((𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁})) → (𝑎 ∈ (𝑉 ∖ {𝑁}) ∧ ∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎))))
55	54	reximdv2 3162	. . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑒 ∈ 𝒫 𝑉) → (∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}) → ∃𝑎 ∈ (𝑉 ∖ {𝑁})∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎)))
56	55	expimpd 452	. . . 4 ⊢ (𝑁 ∈ 𝑉 → ((𝑒 ∈ 𝒫 𝑉 ∧ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁})) → ∃𝑎 ∈ (𝑉 ∖ {𝑁})∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎)))
57		eqeq1 2734	. . . . . . 7 ⊢ (𝑥 = 𝑒 → (𝑥 = {𝑎, 𝑁} ↔ 𝑒 = {𝑎, 𝑁}))
58	57	anbi2d 627	. . . . . 6 ⊢ (𝑥 = 𝑒 → ((𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}) ↔ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁})))
59	58	rexbidv 3176	. . . . 5 ⊢ (𝑥 = 𝑒 → (∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}) ↔ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁})))
60	59, 18	elrab2 3685	. . . 4 ⊢ (𝑒 ∈ 𝑃 ↔ (𝑒 ∈ 𝒫 𝑉 ∧ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁})))
61		reu6 3721	. . . 4 ⊢ (∃!𝑥 ∈ (𝑉 ∖ {𝑁})𝑒 = {𝑥, 𝑁} ↔ ∃𝑎 ∈ (𝑉 ∖ {𝑁})∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎))
62	56, 60, 61	3imtr4g 295	. . 3 ⊢ (𝑁 ∈ 𝑉 → (𝑒 ∈ 𝑃 → ∃!𝑥 ∈ (𝑉 ∖ {𝑁})𝑒 = {𝑥, 𝑁}))
63	62	ralrimiv 3143	. 2 ⊢ (𝑁 ∈ 𝑉 → ∀𝑒 ∈ 𝑃 ∃!𝑥 ∈ (𝑉 ∖ {𝑁})𝑒 = {𝑥, 𝑁})
64		cusgrfi.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ (𝑉 ∖ {𝑁}) ↦ {𝑥, 𝑁})
65	64	f1ompt 7111	. 2 ⊢ (𝐹:(𝑉 ∖ {𝑁})–1-1-onto→𝑃 ↔ (∀𝑥 ∈ (𝑉 ∖ {𝑁}){𝑥, 𝑁} ∈ 𝑃 ∧ ∀𝑒 ∈ 𝑃 ∃!𝑥 ∈ (𝑉 ∖ {𝑁})𝑒 = {𝑥, 𝑁}))
66	30, 63, 65	sylanbrc 581	1 ⊢ (𝑁 ∈ 𝑉 → 𝐹:(𝑉 ∖ {𝑁})–1-1-onto→𝑃)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104   ≠ wne 2938  ∀wral 3059  ∃wrex 3068  ∃!wreu 3372  {crab 3430   ∖ cdif 3944  𝒫 cpw 4601  {csn 4627  {cpr 4629   ↦ cmpt 5230  –1-1-onto→wf1o 6541  ‘cfv 6542  Vtxcvtx 28523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  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-br 5148  df-opab 5210  df-mpt 5231  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-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549

This theorem is referenced by:  cusgrfilem3  28981