Theorem kqt0lem 23723

Description: Lemma for kqt0 23733. (Contributed by Mario Carneiro, 23-Mar-2015.)

Hypothesis

Ref	Expression
kqval.2	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})

Assertion

Ref	Expression
kqt0lem	⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ Kol2)

Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝑋,𝑦

Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem kqt0lem

Dummy variables 𝑤 𝑧 𝑎 𝑏 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		kqval.2	. . . . . . . . . 10 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
2	1	kqopn 23721	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽) → (𝐹 “ 𝑤) ∈ (KQ‘𝐽))
3	2	adantlr 722	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ 𝑤 ∈ 𝐽) → (𝐹 “ 𝑤) ∈ (KQ‘𝐽))
4		eleq2 2830	. . . . . . . . . 10 ⊢ (𝑧 = (𝐹 “ 𝑤) → ((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑎) ∈ (𝐹 “ 𝑤)))
5		eleq2 2830	. . . . . . . . . 10 ⊢ (𝑧 = (𝐹 “ 𝑤) → ((𝐹‘𝑏) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ (𝐹 “ 𝑤)))
6	4, 5	bibi12d 347	. . . . . . . . 9 ⊢ (𝑧 = (𝐹 “ 𝑤) → (((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧) ↔ ((𝐹‘𝑎) ∈ (𝐹 “ 𝑤) ↔ (𝐹‘𝑏) ∈ (𝐹 “ 𝑤))))
7	6	rspcv 3558	. . . . . . . 8 ⊢ ((𝐹 “ 𝑤) ∈ (KQ‘𝐽) → (∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧) → ((𝐹‘𝑎) ∈ (𝐹 “ 𝑤) ↔ (𝐹‘𝑏) ∈ (𝐹 “ 𝑤))))
8	3, 7	syl 17	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ 𝑤 ∈ 𝐽) → (∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧) → ((𝐹‘𝑎) ∈ (𝐹 “ 𝑤) ↔ (𝐹‘𝑏) ∈ (𝐹 “ 𝑤))))
9	1	kqfvima 23717	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽 ∧ 𝑎 ∈ 𝑋) → (𝑎 ∈ 𝑤 ↔ (𝐹‘𝑎) ∈ (𝐹 “ 𝑤)))
10	9	3expa 1125	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽) ∧ 𝑎 ∈ 𝑋) → (𝑎 ∈ 𝑤 ↔ (𝐹‘𝑎) ∈ (𝐹 “ 𝑤)))
11	10	adantrr 724	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎 ∈ 𝑤 ↔ (𝐹‘𝑎) ∈ (𝐹 “ 𝑤)))
12	1	kqfvima 23717	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽 ∧ 𝑏 ∈ 𝑋) → (𝑏 ∈ 𝑤 ↔ (𝐹‘𝑏) ∈ (𝐹 “ 𝑤)))
13	12	3expa 1125	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽) ∧ 𝑏 ∈ 𝑋) → (𝑏 ∈ 𝑤 ↔ (𝐹‘𝑏) ∈ (𝐹 “ 𝑤)))
14	13	adantrl 723	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑏 ∈ 𝑤 ↔ (𝐹‘𝑏) ∈ (𝐹 “ 𝑤)))
15	11, 14	bibi12d 347	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝑎 ∈ 𝑤 ↔ 𝑏 ∈ 𝑤) ↔ ((𝐹‘𝑎) ∈ (𝐹 “ 𝑤) ↔ (𝐹‘𝑏) ∈ (𝐹 “ 𝑤))))
16	15	an32s 659	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ 𝑤 ∈ 𝐽) → ((𝑎 ∈ 𝑤 ↔ 𝑏 ∈ 𝑤) ↔ ((𝐹‘𝑎) ∈ (𝐹 “ 𝑤) ↔ (𝐹‘𝑏) ∈ (𝐹 “ 𝑤))))
17	8, 16	sylibrd 261	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ 𝑤 ∈ 𝐽) → (∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧) → (𝑎 ∈ 𝑤 ↔ 𝑏 ∈ 𝑤)))
18	17	ralrimdva 3141	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧) → ∀𝑤 ∈ 𝐽 (𝑎 ∈ 𝑤 ↔ 𝑏 ∈ 𝑤)))
19	1	kqfeq 23711	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ ∀𝑦 ∈ 𝐽 (𝑎 ∈ 𝑦 ↔ 𝑏 ∈ 𝑦)))
20	19	3expb 1127	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ ∀𝑦 ∈ 𝐽 (𝑎 ∈ 𝑦 ↔ 𝑏 ∈ 𝑦)))
21		elequ2 2136	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (𝑎 ∈ 𝑦 ↔ 𝑎 ∈ 𝑤))
22		elequ2 2136	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (𝑏 ∈ 𝑦 ↔ 𝑏 ∈ 𝑤))
23	21, 22	bibi12d 347	. . . . . . 7 ⊢ (𝑦 = 𝑤 → ((𝑎 ∈ 𝑦 ↔ 𝑏 ∈ 𝑦) ↔ (𝑎 ∈ 𝑤 ↔ 𝑏 ∈ 𝑤)))
24	23	cbvralvw 3219	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐽 (𝑎 ∈ 𝑦 ↔ 𝑏 ∈ 𝑦) ↔ ∀𝑤 ∈ 𝐽 (𝑎 ∈ 𝑤 ↔ 𝑏 ∈ 𝑤))
25	20, 24	bitrdi 289	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ ∀𝑤 ∈ 𝐽 (𝑎 ∈ 𝑤 ↔ 𝑏 ∈ 𝑤)))
26	18, 25	sylibrd 261	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧) → (𝐹‘𝑎) = (𝐹‘𝑏)))
27	26	ralrimivva 3184	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧) → (𝐹‘𝑎) = (𝐹‘𝑏)))
28	1	kqffn 23712	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
29		eleq1 2829	. . . . . . . . . 10 ⊢ (𝑢 = (𝐹‘𝑎) → (𝑢 ∈ 𝑧 ↔ (𝐹‘𝑎) ∈ 𝑧))
30	29	bibi1d 345	. . . . . . . . 9 ⊢ (𝑢 = (𝐹‘𝑎) → ((𝑢 ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) ↔ ((𝐹‘𝑎) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧)))
31	30	ralbidv 3164	. . . . . . . 8 ⊢ (𝑢 = (𝐹‘𝑎) → (∀𝑧 ∈ (KQ‘𝐽)(𝑢 ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) ↔ ∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧)))
32		eqeq1 2745	. . . . . . . 8 ⊢ (𝑢 = (𝐹‘𝑎) → (𝑢 = 𝑣 ↔ (𝐹‘𝑎) = 𝑣))
33	31, 32	imbi12d 346	. . . . . . 7 ⊢ (𝑢 = (𝐹‘𝑎) → ((∀𝑧 ∈ (KQ‘𝐽)(𝑢 ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) → 𝑢 = 𝑣) ↔ (∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) → (𝐹‘𝑎) = 𝑣)))
34	33	ralbidv 3164	. . . . . 6 ⊢ (𝑢 = (𝐹‘𝑎) → (∀𝑣 ∈ ran 𝐹(∀𝑧 ∈ (KQ‘𝐽)(𝑢 ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) → 𝑢 = 𝑣) ↔ ∀𝑣 ∈ ran 𝐹(∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) → (𝐹‘𝑎) = 𝑣)))
35	34	ralrn 7033	. . . . 5 ⊢ (𝐹 Fn 𝑋 → (∀𝑢 ∈ ran 𝐹∀𝑣 ∈ ran 𝐹(∀𝑧 ∈ (KQ‘𝐽)(𝑢 ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) → 𝑢 = 𝑣) ↔ ∀𝑎 ∈ 𝑋 ∀𝑣 ∈ ran 𝐹(∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) → (𝐹‘𝑎) = 𝑣)))
36		eleq1 2829	. . . . . . . . . 10 ⊢ (𝑣 = (𝐹‘𝑏) → (𝑣 ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧))
37	36	bibi2d 344	. . . . . . . . 9 ⊢ (𝑣 = (𝐹‘𝑏) → (((𝐹‘𝑎) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) ↔ ((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧)))
38	37	ralbidv 3164	. . . . . . . 8 ⊢ (𝑣 = (𝐹‘𝑏) → (∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) ↔ ∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧)))
39		eqeq2 2753	. . . . . . . 8 ⊢ (𝑣 = (𝐹‘𝑏) → ((𝐹‘𝑎) = 𝑣 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)))
40	38, 39	imbi12d 346	. . . . . . 7 ⊢ (𝑣 = (𝐹‘𝑏) → ((∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) → (𝐹‘𝑎) = 𝑣) ↔ (∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧) → (𝐹‘𝑎) = (𝐹‘𝑏))))
41	40	ralrn 7033	. . . . . 6 ⊢ (𝐹 Fn 𝑋 → (∀𝑣 ∈ ran 𝐹(∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) → (𝐹‘𝑎) = 𝑣) ↔ ∀𝑏 ∈ 𝑋 (∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧) → (𝐹‘𝑎) = (𝐹‘𝑏))))
42	41	ralbidv 3164	. . . . 5 ⊢ (𝐹 Fn 𝑋 → (∀𝑎 ∈ 𝑋 ∀𝑣 ∈ ran 𝐹(∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) → (𝐹‘𝑎) = 𝑣) ↔ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧) → (𝐹‘𝑎) = (𝐹‘𝑏))))
43	35, 42	bitrd 281	. . . 4 ⊢ (𝐹 Fn 𝑋 → (∀𝑢 ∈ ran 𝐹∀𝑣 ∈ ran 𝐹(∀𝑧 ∈ (KQ‘𝐽)(𝑢 ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) → 𝑢 = 𝑣) ↔ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧) → (𝐹‘𝑎) = (𝐹‘𝑏))))
44	28, 43	syl 17	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑢 ∈ ran 𝐹∀𝑣 ∈ ran 𝐹(∀𝑧 ∈ (KQ‘𝐽)(𝑢 ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) → 𝑢 = 𝑣) ↔ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧) → (𝐹‘𝑎) = (𝐹‘𝑏))))
45	27, 44	mpbird 259	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∀𝑢 ∈ ran 𝐹∀𝑣 ∈ ran 𝐹(∀𝑧 ∈ (KQ‘𝐽)(𝑢 ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) → 𝑢 = 𝑣))
46	1	kqtopon 23714	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
47		ist0-2 23331	. . 3 ⊢ ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → ((KQ‘𝐽) ∈ Kol2 ↔ ∀𝑢 ∈ ran 𝐹∀𝑣 ∈ ran 𝐹(∀𝑧 ∈ (KQ‘𝐽)(𝑢 ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) → 𝑢 = 𝑣)))
48	46, 47	syl 17	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ((KQ‘𝐽) ∈ Kol2 ↔ ∀𝑢 ∈ ran 𝐹∀𝑣 ∈ ran 𝐹(∀𝑧 ∈ (KQ‘𝐽)(𝑢 ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) → 𝑢 = 𝑣)))
49	45, 48	mpbird 259	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ Kol2)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121  ∀wral 3055  {crab 3393   ↦ cmpt 5156  ran crn 5622   “ cima 5624   Fn wfn 6484  ‘cfv 6489  TopOnctopon 22897  Kol2ct0 23293  KQckq 23680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7363  df-oprab 7364  df-mpo 7365  df-qtop 17466  df-top 22881  df-topon 22898  df-t0 23300  df-kq 23681

This theorem is referenced by:  kqt0  23733  t0kq  23805