Theorem cusgrfilem2 28404

Description: Lemma 2 for cusgrfi 28406. (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 4086	. . . . 5 ⊢ (𝑥 ∈ (𝑉 ∖ {𝑁}) → 𝑥 ∈ 𝑉)
2		id 22	. . . . 5 ⊢ (𝑁 ∈ 𝑉 → 𝑁 ∈ 𝑉)
3		prelpwi 5404	. . . . 5 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) → {𝑥, 𝑁} ∈ 𝒫 𝑉)
4	1, 2, 3	syl2anr 597	. . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → {𝑥, 𝑁} ∈ 𝒫 𝑉)
5	1	adantl 482	. . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → 𝑥 ∈ 𝑉)
6		eldifsni 4750	. . . . . . 7 ⊢ (𝑥 ∈ (𝑉 ∖ {𝑁}) → 𝑥 ≠ 𝑁)
7	6	adantl 482	. . . . . 6 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → 𝑥 ≠ 𝑁)
8		eqidd 2737	. . . . . 6 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → {𝑥, 𝑁} = {𝑥, 𝑁})
9	7, 8	jca 512	. . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑥 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑥, 𝑁}))
10		id 22	. . . . . 6 ⊢ (𝑥 ∈ 𝑉 → 𝑥 ∈ 𝑉)
11		neeq1 3006	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → (𝑎 ≠ 𝑁 ↔ 𝑥 ≠ 𝑁))
12		preq1 4694	. . . . . . . . 9 ⊢ (𝑎 = 𝑥 → {𝑎, 𝑁} = {𝑥, 𝑁})
13	12	eqeq2d 2747	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → ({𝑥, 𝑁} = {𝑎, 𝑁} ↔ {𝑥, 𝑁} = {𝑥, 𝑁}))
14	11, 13	anbi12d 631	. . . . . . 7 ⊢ (𝑎 = 𝑥 → ((𝑎 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁}) ↔ (𝑥 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑥, 𝑁})))
15	14	adantl 482	. . . . . 6 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑎 = 𝑥) → ((𝑎 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁}) ↔ (𝑥 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑥, 𝑁})))
16	10, 15	rspcedv 3574	. . . . 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 2829	. . . . 5 ⊢ ({𝑥, 𝑁} ∈ 𝑃 ↔ {𝑥, 𝑁} ∈ {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁})})
20		eqeq1 2740	. . . . . . . 8 ⊢ (𝑣 = {𝑥, 𝑁} → (𝑣 = {𝑎, 𝑁} ↔ {𝑥, 𝑁} = {𝑎, 𝑁}))
21	20	anbi2d 629	. . . . . . 7 ⊢ (𝑣 = {𝑥, 𝑁} → ((𝑎 ≠ 𝑁 ∧ 𝑣 = {𝑎, 𝑁}) ↔ (𝑎 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁})))
22	21	rexbidv 3175	. . . . . 6 ⊢ (𝑣 = {𝑥, 𝑁} → (∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑣 = {𝑎, 𝑁}) ↔ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁})))
23		eqeq1 2740	. . . . . . . . 9 ⊢ (𝑥 = 𝑣 → (𝑥 = {𝑎, 𝑁} ↔ 𝑣 = {𝑎, 𝑁}))
24	23	anbi2d 629	. . . . . . . 8 ⊢ (𝑥 = 𝑣 → ((𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}) ↔ (𝑎 ≠ 𝑁 ∧ 𝑣 = {𝑎, 𝑁})))
25	24	rexbidv 3175	. . . . . . 7 ⊢ (𝑥 = 𝑣 → (∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}) ↔ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑣 = {𝑎, 𝑁})))
26	25	cbvrabv 3417	. . . . . 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 3648	. . . . 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 583	. . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → {𝑥, 𝑁} ∈ 𝑃)
30	29	ralrimiva 3143	. 2 ⊢ (𝑁 ∈ 𝑉 → ∀𝑥 ∈ (𝑉 ∖ {𝑁}){𝑥, 𝑁} ∈ 𝑃)
31		simpl 483	. . . . . . . . . . 11 ⊢ ((𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}) → 𝑎 ≠ 𝑁)
32	31	anim2i 617	. . . . . . . . . 10 ⊢ ((𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁})) → (𝑎 ∈ 𝑉 ∧ 𝑎 ≠ 𝑁))
33	32	adantl 482	. . . . . . . . 9 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝑒 ∈ 𝒫 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}))) → (𝑎 ∈ 𝑉 ∧ 𝑎 ≠ 𝑁))
34		eldifsn 4747	. . . . . . . . 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 2740	. . . . . . . . . . . . . 14 ⊢ (𝑒 = {𝑎, 𝑁} → (𝑒 = {𝑥, 𝑁} ↔ {𝑎, 𝑁} = {𝑥, 𝑁}))
37	36	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}) → (𝑒 = {𝑥, 𝑁} ↔ {𝑎, 𝑁} = {𝑥, 𝑁}))
38	37	ad2antlr 725	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁})) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑒 = {𝑥, 𝑁} ↔ {𝑎, 𝑁} = {𝑥, 𝑁}))
39		vex 3449	. . . . . . . . . . . . . 14 ⊢ 𝑎 ∈ V
40		vex 3449	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
41	39, 40	preqr1 4806	. . . . . . . . . . . . 13 ⊢ ({𝑎, 𝑁} = {𝑥, 𝑁} → 𝑎 = 𝑥)
42	41	equcomd 2022	. . . . . . . . . . . 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 712	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ 𝑉 ∧ 𝑒 ∈ 𝒫 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}))) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑒 = {𝑥, 𝑁} → 𝑥 = 𝑎))
45	12	equcoms 2023	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑎 → {𝑎, 𝑁} = {𝑥, 𝑁})
46	45	eqeq2d 2747	. . . . . . . . . . . . . 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 482	. . . . . . . . . . . 12 ⊢ ((𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}) → (𝑥 = 𝑎 → 𝑒 = {𝑥, 𝑁}))
49	48	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁})) → (𝑥 = 𝑎 → 𝑒 = {𝑥, 𝑁}))
50	49	ad2antlr 725	. . . . . . . . . 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 3143	. . . . . . . 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 512	. . . . . . 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 413	. . . . . 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 3161	. . . . 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 454	. . . 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 2740	. . . . . . 7 ⊢ (𝑥 = 𝑒 → (𝑥 = {𝑎, 𝑁} ↔ 𝑒 = {𝑎, 𝑁}))
58	57	anbi2d 629	. . . . . 6 ⊢ (𝑥 = 𝑒 → ((𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}) ↔ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁})))
59	58	rexbidv 3175	. . . . 5 ⊢ (𝑥 = 𝑒 → (∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}) ↔ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁})))
60	59, 18	elrab2 3648	. . . 4 ⊢ (𝑒 ∈ 𝑃 ↔ (𝑒 ∈ 𝒫 𝑉 ∧ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁})))
61		reu6 3684	. . . 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 3142	. 2 ⊢ (𝑁 ∈ 𝑉 → ∀𝑒 ∈ 𝑃 ∃!𝑥 ∈ (𝑉 ∖ {𝑁})𝑒 = {𝑥, 𝑁})
64		cusgrfi.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ (𝑉 ∖ {𝑁}) ↦ {𝑥, 𝑁})
65	64	f1ompt 7059	. 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 583	1 ⊢ (𝑁 ∈ 𝑉 → 𝐹:(𝑉 ∖ {𝑁})–1-1-onto→𝑃)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3073  ∃!wreu 3351  {crab 3407   ∖ cdif 3907  𝒫 cpw 4560  {csn 4586  {cpr 4588   ↦ cmpt 5188  –1-1-onto→wf1o 6495  ‘cfv 6496  Vtxcvtx 27947

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 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384

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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503

This theorem is referenced by:  cusgrfilem3  28405