Theorem cusgrafilem2 26008

Description: Lemma 2 for cusgrafi 26010. (Contributed by Alexander van der Vekens, 13-Jan-2018.)

Hypotheses

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

Assertion

Ref	Expression
cusgrafilem2	⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉) → 𝐹:(𝑉 ∖ {𝑁})–1-1-onto→𝑃)

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

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

Proof of Theorem cusgrafilem2

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

Step	Hyp	Ref	Expression
1		eldifsn 4260	. . . . . . 7 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) ↔ (𝑣 ∈ 𝑉 ∧ 𝑣 ≠ 𝑁))
2		simpl 472	. . . . . . 7 ⊢ ((𝑣 ∈ 𝑉 ∧ 𝑣 ≠ 𝑁) → 𝑣 ∈ 𝑉)
3	1, 2	sylbi 206	. . . . . 6 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → 𝑣 ∈ 𝑉)
4		simpr 476	. . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉) → 𝑁 ∈ 𝑉)
5		prelpwi 4842	. . . . . 6 ⊢ ((𝑣 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) → {𝑣, 𝑁} ∈ 𝒫 𝑉)
6	3, 4, 5	syl2anr 494	. . . . 5 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → {𝑣, 𝑁} ∈ 𝒫 𝑉)
7	1	biimpi 205	. . . . . . 7 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → (𝑣 ∈ 𝑉 ∧ 𝑣 ≠ 𝑁))
8	7	adantl 481	. . . . . 6 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → (𝑣 ∈ 𝑉 ∧ 𝑣 ≠ 𝑁))
9		simpr 476	. . . . . . . . 9 ⊢ ((𝑣 ∈ 𝑉 ∧ 𝑣 ≠ 𝑁) → 𝑣 ≠ 𝑁)
10	1, 9	sylbi 206	. . . . . . . 8 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → 𝑣 ≠ 𝑁)
11	10	adantl 481	. . . . . . 7 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → 𝑣 ≠ 𝑁)
12		eqidd 2611	. . . . . . 7 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → {𝑣, 𝑁} = {𝑣, 𝑁})
13	11, 12	jca 553	. . . . . 6 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → (𝑣 ≠ 𝑁 ∧ {𝑣, 𝑁} = {𝑣, 𝑁}))
14		neeq1 2844	. . . . . . . . 9 ⊢ (𝑎 = 𝑣 → (𝑎 ≠ 𝑁 ↔ 𝑣 ≠ 𝑁))
15		preq1 4212	. . . . . . . . . 10 ⊢ (𝑎 = 𝑣 → {𝑎, 𝑁} = {𝑣, 𝑁})
16	15	eqeq2d 2620	. . . . . . . . 9 ⊢ (𝑎 = 𝑣 → ({𝑣, 𝑁} = {𝑎, 𝑁} ↔ {𝑣, 𝑁} = {𝑣, 𝑁}))
17	14, 16	anbi12d 743	. . . . . . . 8 ⊢ (𝑎 = 𝑣 → ((𝑎 ≠ 𝑁 ∧ {𝑣, 𝑁} = {𝑎, 𝑁}) ↔ (𝑣 ≠ 𝑁 ∧ {𝑣, 𝑁} = {𝑣, 𝑁})))
18	17	adantl 481	. . . . . . 7 ⊢ (((𝑣 ∈ 𝑉 ∧ 𝑣 ≠ 𝑁) ∧ 𝑎 = 𝑣) → ((𝑎 ≠ 𝑁 ∧ {𝑣, 𝑁} = {𝑎, 𝑁}) ↔ (𝑣 ≠ 𝑁 ∧ {𝑣, 𝑁} = {𝑣, 𝑁})))
19	2, 18	rspcedv 3286	. . . . . 6 ⊢ ((𝑣 ∈ 𝑉 ∧ 𝑣 ≠ 𝑁) → ((𝑣 ≠ 𝑁 ∧ {𝑣, 𝑁} = {𝑣, 𝑁}) → ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ {𝑣, 𝑁} = {𝑎, 𝑁})))
20	8, 13, 19	sylc 63	. . . . 5 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ {𝑣, 𝑁} = {𝑎, 𝑁}))
21		eqeq1 2614	. . . . . . . 8 ⊢ (𝑥 = {𝑣, 𝑁} → (𝑥 = {𝑎, 𝑁} ↔ {𝑣, 𝑁} = {𝑎, 𝑁}))
22	21	anbi2d 736	. . . . . . 7 ⊢ (𝑥 = {𝑣, 𝑁} → ((𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}) ↔ (𝑎 ≠ 𝑁 ∧ {𝑣, 𝑁} = {𝑎, 𝑁})))
23	22	rexbidv 3034	. . . . . 6 ⊢ (𝑥 = {𝑣, 𝑁} → (∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}) ↔ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ {𝑣, 𝑁} = {𝑎, 𝑁})))
24		cusgrafi.p	. . . . . 6 ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁})}
25	23, 24	elrab2 3333	. . . . 5 ⊢ ({𝑣, 𝑁} ∈ 𝑃 ↔ ({𝑣, 𝑁} ∈ 𝒫 𝑉 ∧ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ {𝑣, 𝑁} = {𝑎, 𝑁})))
26	6, 20, 25	sylanbrc 695	. . . 4 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → {𝑣, 𝑁} ∈ 𝑃)
27	26	ralrimiva 2949	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉) → ∀𝑣 ∈ (𝑉 ∖ {𝑁}){𝑣, 𝑁} ∈ 𝑃)
28		preq1 4212	. . . . 5 ⊢ (𝑥 = 𝑣 → {𝑥, 𝑁} = {𝑣, 𝑁})
29	28	eleq1d 2672	. . . 4 ⊢ (𝑥 = 𝑣 → ({𝑥, 𝑁} ∈ 𝑃 ↔ {𝑣, 𝑁} ∈ 𝑃))
30	29	cbvralv 3147	. . 3 ⊢ (∀𝑥 ∈ (𝑉 ∖ {𝑁}){𝑥, 𝑁} ∈ 𝑃 ↔ ∀𝑣 ∈ (𝑉 ∖ {𝑁}){𝑣, 𝑁} ∈ 𝑃)
31	27, 30	sylibr 223	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉) → ∀𝑥 ∈ (𝑉 ∖ {𝑁}){𝑥, 𝑁} ∈ 𝑃)
32		simpl 472	. . . . . . . . . . 11 ⊢ ((𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}) → 𝑎 ≠ 𝑁)
33	32	anim2i 591	. . . . . . . . . 10 ⊢ ((𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁})) → (𝑎 ∈ 𝑉 ∧ 𝑎 ≠ 𝑁))
34	33	adantl 481	. . . . . . . . 9 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝒫 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}))) → (𝑎 ∈ 𝑉 ∧ 𝑎 ≠ 𝑁))
35		eldifsn 4260	. . . . . . . . 9 ⊢ (𝑎 ∈ (𝑉 ∖ {𝑁}) ↔ (𝑎 ∈ 𝑉 ∧ 𝑎 ≠ 𝑁))
36	34, 35	sylibr 223	. . . . . . . 8 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝒫 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}))) → 𝑎 ∈ (𝑉 ∖ {𝑁}))
37		eqeq1 2614	. . . . . . . . . . . . . 14 ⊢ (𝑒 = {𝑎, 𝑁} → (𝑒 = {𝑥, 𝑁} ↔ {𝑎, 𝑁} = {𝑥, 𝑁}))
38	37	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}) → (𝑒 = {𝑥, 𝑁} ↔ {𝑎, 𝑁} = {𝑥, 𝑁}))
39	38	ad2antlr 759	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁})) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑒 = {𝑥, 𝑁} ↔ {𝑎, 𝑁} = {𝑥, 𝑁}))
40		vex 3176	. . . . . . . . . . . . . 14 ⊢ 𝑎 ∈ V
41		vex 3176	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
42	40, 41	preqr1 4319	. . . . . . . . . . . . 13 ⊢ ({𝑎, 𝑁} = {𝑥, 𝑁} → 𝑎 = 𝑥)
43	42	equcomd 1933	. . . . . . . . . . . 12 ⊢ ({𝑎, 𝑁} = {𝑥, 𝑁} → 𝑥 = 𝑎)
44	39, 43	syl6bi 242	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁})) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑒 = {𝑥, 𝑁} → 𝑥 = 𝑎))
45	44	adantll 746	. . . . . . . . . 10 ⊢ (((((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝒫 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}))) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑒 = {𝑥, 𝑁} → 𝑥 = 𝑎))
46		preq1 4212	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑥 → {𝑎, 𝑁} = {𝑥, 𝑁})
47	46	equcoms 1934	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑎 → {𝑎, 𝑁} = {𝑥, 𝑁})
48	47	eqeq2d 2620	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑎 → (𝑒 = {𝑎, 𝑁} ↔ 𝑒 = {𝑥, 𝑁}))
49	48	biimpcd 238	. . . . . . . . . . . . 13 ⊢ (𝑒 = {𝑎, 𝑁} → (𝑥 = 𝑎 → 𝑒 = {𝑥, 𝑁}))
50	49	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}) → (𝑥 = 𝑎 → 𝑒 = {𝑥, 𝑁}))
51	50	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁})) → (𝑥 = 𝑎 → 𝑒 = {𝑥, 𝑁}))
52	51	ad2antlr 759	. . . . . . . . . 10 ⊢ (((((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝒫 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}))) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑥 = 𝑎 → 𝑒 = {𝑥, 𝑁}))
53	45, 52	impbid 201	. . . . . . . . 9 ⊢ (((((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝒫 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}))) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎))
54	53	ralrimiva 2949	. . . . . . . 8 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝒫 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}))) → ∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎))
55	36, 54	jca 553	. . . . . . 7 ⊢ ((((𝑉 ∈ 𝑊 ∧ 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	ex 449	. . . . . 6 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝒫 𝑉) → ((𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁})) → (𝑎 ∈ (𝑉 ∖ {𝑁}) ∧ ∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎))))
57	56	reximdv2 2997	. . . . 5 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝒫 𝑉) → (∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}) → ∃𝑎 ∈ (𝑉 ∖ {𝑁})∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎)))
58	57	expimpd 627	. . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉) → ((𝑒 ∈ 𝒫 𝑉 ∧ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁})) → ∃𝑎 ∈ (𝑉 ∖ {𝑁})∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎)))
59		eqeq1 2614	. . . . . . 7 ⊢ (𝑥 = 𝑒 → (𝑥 = {𝑎, 𝑁} ↔ 𝑒 = {𝑎, 𝑁}))
60	59	anbi2d 736	. . . . . 6 ⊢ (𝑥 = 𝑒 → ((𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}) ↔ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁})))
61	60	rexbidv 3034	. . . . 5 ⊢ (𝑥 = 𝑒 → (∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}) ↔ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁})))
62	61, 24	elrab2 3333	. . . 4 ⊢ (𝑒 ∈ 𝑃 ↔ (𝑒 ∈ 𝒫 𝑉 ∧ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁})))
63		reu6 3362	. . . 4 ⊢ (∃!𝑥 ∈ (𝑉 ∖ {𝑁})𝑒 = {𝑥, 𝑁} ↔ ∃𝑎 ∈ (𝑉 ∖ {𝑁})∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎))
64	58, 62, 63	3imtr4g 284	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉) → (𝑒 ∈ 𝑃 → ∃!𝑥 ∈ (𝑉 ∖ {𝑁})𝑒 = {𝑥, 𝑁}))
65	64	ralrimiv 2948	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉) → ∀𝑒 ∈ 𝑃 ∃!𝑥 ∈ (𝑉 ∖ {𝑁})𝑒 = {𝑥, 𝑁})
66		cusgrafi.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ (𝑉 ∖ {𝑁}) ↦ {𝑥, 𝑁})
67	66	f1ompt 6290	. 2 ⊢ (𝐹:(𝑉 ∖ {𝑁})–1-1-onto→𝑃 ↔ (∀𝑥 ∈ (𝑉 ∖ {𝑁}){𝑥, 𝑁} ∈ 𝑃 ∧ ∀𝑒 ∈ 𝑃 ∃!𝑥 ∈ (𝑉 ∖ {𝑁})𝑒 = {𝑥, 𝑁}))
68	31, 65, 67	sylanbrc 695	1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉) → 𝐹:(𝑉 ∖ {𝑁})–1-1-onto→𝑃)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898  {crab 2900   ∖ cdif 3537  𝒫 cpw 4108  {csn 4125  {cpr 4127   ↦ cmpt 4643  –1-1-onto→wf1o 5803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812

This theorem is referenced by:  cusgrafilem3  26009