Theorem cusgrfilem2 29436

Description: Lemma 2 for cusgrfi 29438. (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 4106	. . . . 5 ⊢ (𝑥 ∈ (𝑉 ∖ {𝑁}) → 𝑥 ∈ 𝑉)
2		id 22	. . . . 5 ⊢ (𝑁 ∈ 𝑉 → 𝑁 ∈ 𝑉)
3		prelpwi 5422	. . . . 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 481	. . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → 𝑥 ∈ 𝑉)
6		eldifsni 4766	. . . . . . 7 ⊢ (𝑥 ∈ (𝑉 ∖ {𝑁}) → 𝑥 ≠ 𝑁)
7	6	adantl 481	. . . . . 6 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → 𝑥 ≠ 𝑁)
8		eqidd 2736	. . . . . 6 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → {𝑥, 𝑁} = {𝑥, 𝑁})
9	7, 8	jca 511	. . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑥 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑥, 𝑁}))
10		id 22	. . . . . 6 ⊢ (𝑥 ∈ 𝑉 → 𝑥 ∈ 𝑉)
11		neeq1 2994	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → (𝑎 ≠ 𝑁 ↔ 𝑥 ≠ 𝑁))
12		preq1 4709	. . . . . . . . 9 ⊢ (𝑎 = 𝑥 → {𝑎, 𝑁} = {𝑥, 𝑁})
13	12	eqeq2d 2746	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → ({𝑥, 𝑁} = {𝑎, 𝑁} ↔ {𝑥, 𝑁} = {𝑥, 𝑁}))
14	11, 13	anbi12d 632	. . . . . . 7 ⊢ (𝑎 = 𝑥 → ((𝑎 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁}) ↔ (𝑥 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑥, 𝑁})))
15	14	adantl 481	. . . . . 6 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑎 = 𝑥) → ((𝑎 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁}) ↔ (𝑥 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑥, 𝑁})))
16	10, 15	rspcedv 3594	. . . . 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 2826	. . . . 5 ⊢ ({𝑥, 𝑁} ∈ 𝑃 ↔ {𝑥, 𝑁} ∈ {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁})})
20		eqeq1 2739	. . . . . . . 8 ⊢ (𝑣 = {𝑥, 𝑁} → (𝑣 = {𝑎, 𝑁} ↔ {𝑥, 𝑁} = {𝑎, 𝑁}))
21	20	anbi2d 630	. . . . . . 7 ⊢ (𝑣 = {𝑥, 𝑁} → ((𝑎 ≠ 𝑁 ∧ 𝑣 = {𝑎, 𝑁}) ↔ (𝑎 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁})))
22	21	rexbidv 3164	. . . . . 6 ⊢ (𝑣 = {𝑥, 𝑁} → (∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑣 = {𝑎, 𝑁}) ↔ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁})))
23		eqeq1 2739	. . . . . . . . 9 ⊢ (𝑥 = 𝑣 → (𝑥 = {𝑎, 𝑁} ↔ 𝑣 = {𝑎, 𝑁}))
24	23	anbi2d 630	. . . . . . . 8 ⊢ (𝑥 = 𝑣 → ((𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}) ↔ (𝑎 ≠ 𝑁 ∧ 𝑣 = {𝑎, 𝑁})))
25	24	rexbidv 3164	. . . . . . 7 ⊢ (𝑥 = 𝑣 → (∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}) ↔ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑣 = {𝑎, 𝑁})))
26	25	cbvrabv 3426	. . . . . 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 3674	. . . . 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 275	. . . 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 3132	. 2 ⊢ (𝑁 ∈ 𝑉 → ∀𝑥 ∈ (𝑉 ∖ {𝑁}){𝑥, 𝑁} ∈ 𝑃)
31		simpl 482	. . . . . . . . . . 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 481	. . . . . . . . 9 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝑒 ∈ 𝒫 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}))) → (𝑎 ∈ 𝑉 ∧ 𝑎 ≠ 𝑁))
34		eldifsn 4762	. . . . . . . . 9 ⊢ (𝑎 ∈ (𝑉 ∖ {𝑁}) ↔ (𝑎 ∈ 𝑉 ∧ 𝑎 ≠ 𝑁))
35	33, 34	sylibr 234	. . . . . . . 8 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝑒 ∈ 𝒫 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}))) → 𝑎 ∈ (𝑉 ∖ {𝑁}))
36		eqeq1 2739	. . . . . . . . . . . . . 14 ⊢ (𝑒 = {𝑎, 𝑁} → (𝑒 = {𝑥, 𝑁} ↔ {𝑎, 𝑁} = {𝑥, 𝑁}))
37	36	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}) → (𝑒 = {𝑥, 𝑁} ↔ {𝑎, 𝑁} = {𝑥, 𝑁}))
38	37	ad2antlr 727	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁})) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑒 = {𝑥, 𝑁} ↔ {𝑎, 𝑁} = {𝑥, 𝑁}))
39		vex 3463	. . . . . . . . . . . . . 14 ⊢ 𝑎 ∈ V
40		vex 3463	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
41	39, 40	preqr1 4824	. . . . . . . . . . . . 13 ⊢ ({𝑎, 𝑁} = {𝑥, 𝑁} → 𝑎 = 𝑥)
42	41	equcomd 2018	. . . . . . . . . . . 12 ⊢ ({𝑎, 𝑁} = {𝑥, 𝑁} → 𝑥 = 𝑎)
43	38, 42	biimtrdi 253	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁})) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑒 = {𝑥, 𝑁} → 𝑥 = 𝑎))
44	43	adantll 714	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ 𝑉 ∧ 𝑒 ∈ 𝒫 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}))) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑒 = {𝑥, 𝑁} → 𝑥 = 𝑎))
45	12	equcoms 2019	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑎 → {𝑎, 𝑁} = {𝑥, 𝑁})
46	45	eqeq2d 2746	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑎 → (𝑒 = {𝑎, 𝑁} ↔ 𝑒 = {𝑥, 𝑁}))
47	46	biimpcd 249	. . . . . . . . . . . . 13 ⊢ (𝑒 = {𝑎, 𝑁} → (𝑥 = 𝑎 → 𝑒 = {𝑥, 𝑁}))
48	47	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}) → (𝑥 = 𝑎 → 𝑒 = {𝑥, 𝑁}))
49	48	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁})) → (𝑥 = 𝑎 → 𝑒 = {𝑥, 𝑁}))
50	49	ad2antlr 727	. . . . . . . . . 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 212	. . . . . . . . 9 ⊢ ((((𝑁 ∈ 𝑉 ∧ 𝑒 ∈ 𝒫 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}))) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎))
52	51	ralrimiva 3132	. . . . . . . 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 511	. . . . . . 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 412	. . . . . 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 3150	. . . . 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 453	. . . 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 2739	. . . . . . 7 ⊢ (𝑥 = 𝑒 → (𝑥 = {𝑎, 𝑁} ↔ 𝑒 = {𝑎, 𝑁}))
58	57	anbi2d 630	. . . . . 6 ⊢ (𝑥 = 𝑒 → ((𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}) ↔ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁})))
59	58	rexbidv 3164	. . . . 5 ⊢ (𝑥 = 𝑒 → (∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}) ↔ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁})))
60	59, 18	elrab2 3674	. . . 4 ⊢ (𝑒 ∈ 𝑃 ↔ (𝑒 ∈ 𝒫 𝑉 ∧ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁})))
61		reu6 3709	. . . 4 ⊢ (∃!𝑥 ∈ (𝑉 ∖ {𝑁})𝑒 = {𝑥, 𝑁} ↔ ∃𝑎 ∈ (𝑉 ∖ {𝑁})∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎))
62	56, 60, 61	3imtr4g 296	. . 3 ⊢ (𝑁 ∈ 𝑉 → (𝑒 ∈ 𝑃 → ∃!𝑥 ∈ (𝑉 ∖ {𝑁})𝑒 = {𝑥, 𝑁}))
63	62	ralrimiv 3131	. 2 ⊢ (𝑁 ∈ 𝑉 → ∀𝑒 ∈ 𝑃 ∃!𝑥 ∈ (𝑉 ∖ {𝑁})𝑒 = {𝑥, 𝑁})
64		cusgrfi.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ (𝑉 ∖ {𝑁}) ↦ {𝑥, 𝑁})
65	64	f1ompt 7101	. 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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  ∀wral 3051  ∃wrex 3060  ∃!wreu 3357  {crab 3415   ∖ cdif 3923  𝒫 cpw 4575  {csn 4601  {cpr 4603   ↦ cmpt 5201  –1-1-onto→wf1o 6530  ‘cfv 6531  Vtxcvtx 28975

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538

This theorem is referenced by:  cusgrfilem3  29437