Theorem kqt0lem 23630

Description: Lemma for kqt0 23640. (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 23628	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽) → (𝐹 “ 𝑤) ∈ (KQ‘𝐽))
3	2	adantlr 715	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ 𝑤 ∈ 𝐽) → (𝐹 “ 𝑤) ∈ (KQ‘𝐽))
4		eleq2 2818	. . . . . . . . . 10 ⊢ (𝑧 = (𝐹 “ 𝑤) → ((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑎) ∈ (𝐹 “ 𝑤)))
5		eleq2 2818	. . . . . . . . . 10 ⊢ (𝑧 = (𝐹 “ 𝑤) → ((𝐹‘𝑏) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ (𝐹 “ 𝑤)))
6	4, 5	bibi12d 345	. . . . . . . . 9 ⊢ (𝑧 = (𝐹 “ 𝑤) → (((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧) ↔ ((𝐹‘𝑎) ∈ (𝐹 “ 𝑤) ↔ (𝐹‘𝑏) ∈ (𝐹 “ 𝑤))))
7	6	rspcv 3587	. . . . . . . 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 23624	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽 ∧ 𝑎 ∈ 𝑋) → (𝑎 ∈ 𝑤 ↔ (𝐹‘𝑎) ∈ (𝐹 “ 𝑤)))
10	9	3expa 1118	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽) ∧ 𝑎 ∈ 𝑋) → (𝑎 ∈ 𝑤 ↔ (𝐹‘𝑎) ∈ (𝐹 “ 𝑤)))
11	10	adantrr 717	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎 ∈ 𝑤 ↔ (𝐹‘𝑎) ∈ (𝐹 “ 𝑤)))
12	1	kqfvima 23624	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽 ∧ 𝑏 ∈ 𝑋) → (𝑏 ∈ 𝑤 ↔ (𝐹‘𝑏) ∈ (𝐹 “ 𝑤)))
13	12	3expa 1118	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽) ∧ 𝑏 ∈ 𝑋) → (𝑏 ∈ 𝑤 ↔ (𝐹‘𝑏) ∈ (𝐹 “ 𝑤)))
14	13	adantrl 716	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑏 ∈ 𝑤 ↔ (𝐹‘𝑏) ∈ (𝐹 “ 𝑤)))
15	11, 14	bibi12d 345	. . . . . . . 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 652	. . . . . . 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 259	. . . . . 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 3134	. . . . 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 23618	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ ∀𝑦 ∈ 𝐽 (𝑎 ∈ 𝑦 ↔ 𝑏 ∈ 𝑦)))
20	19	3expb 1120	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ ∀𝑦 ∈ 𝐽 (𝑎 ∈ 𝑦 ↔ 𝑏 ∈ 𝑦)))
21		elequ2 2124	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (𝑎 ∈ 𝑦 ↔ 𝑎 ∈ 𝑤))
22		elequ2 2124	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (𝑏 ∈ 𝑦 ↔ 𝑏 ∈ 𝑤))
23	21, 22	bibi12d 345	. . . . . . 7 ⊢ (𝑦 = 𝑤 → ((𝑎 ∈ 𝑦 ↔ 𝑏 ∈ 𝑦) ↔ (𝑎 ∈ 𝑤 ↔ 𝑏 ∈ 𝑤)))
24	23	cbvralvw 3216	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐽 (𝑎 ∈ 𝑦 ↔ 𝑏 ∈ 𝑦) ↔ ∀𝑤 ∈ 𝐽 (𝑎 ∈ 𝑤 ↔ 𝑏 ∈ 𝑤))
25	20, 24	bitrdi 287	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ ∀𝑤 ∈ 𝐽 (𝑎 ∈ 𝑤 ↔ 𝑏 ∈ 𝑤)))
26	18, 25	sylibrd 259	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧) → (𝐹‘𝑎) = (𝐹‘𝑏)))
27	26	ralrimivva 3181	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧) → (𝐹‘𝑎) = (𝐹‘𝑏)))
28	1	kqffn 23619	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
29		eleq1 2817	. . . . . . . . . 10 ⊢ (𝑢 = (𝐹‘𝑎) → (𝑢 ∈ 𝑧 ↔ (𝐹‘𝑎) ∈ 𝑧))
30	29	bibi1d 343	. . . . . . . . 9 ⊢ (𝑢 = (𝐹‘𝑎) → ((𝑢 ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) ↔ ((𝐹‘𝑎) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧)))
31	30	ralbidv 3157	. . . . . . . 8 ⊢ (𝑢 = (𝐹‘𝑎) → (∀𝑧 ∈ (KQ‘𝐽)(𝑢 ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) ↔ ∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧)))
32		eqeq1 2734	. . . . . . . 8 ⊢ (𝑢 = (𝐹‘𝑎) → (𝑢 = 𝑣 ↔ (𝐹‘𝑎) = 𝑣))
33	31, 32	imbi12d 344	. . . . . . 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 3157	. . . . . 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 7063	. . . . 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 2817	. . . . . . . . . 10 ⊢ (𝑣 = (𝐹‘𝑏) → (𝑣 ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧))
37	36	bibi2d 342	. . . . . . . . 9 ⊢ (𝑣 = (𝐹‘𝑏) → (((𝐹‘𝑎) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) ↔ ((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧)))
38	37	ralbidv 3157	. . . . . . . 8 ⊢ (𝑣 = (𝐹‘𝑏) → (∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) ↔ ∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧)))
39		eqeq2 2742	. . . . . . . 8 ⊢ (𝑣 = (𝐹‘𝑏) → ((𝐹‘𝑎) = 𝑣 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)))
40	38, 39	imbi12d 344	. . . . . . 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 7063	. . . . . 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 3157	. . . . 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 279	. . . 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 257	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∀𝑢 ∈ ran 𝐹∀𝑣 ∈ ran 𝐹(∀𝑧 ∈ (KQ‘𝐽)(𝑢 ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) → 𝑢 = 𝑣))
46	1	kqtopon 23621	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
47		ist0-2 23238	. . 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 257	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ Kol2)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  {crab 3408   ↦ cmpt 5191  ran crn 5642   “ cima 5644   Fn wfn 6509  ‘cfv 6514  TopOnctopon 22804  Kol2ct0 23200  KQckq 23587

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-qtop 17477  df-top 22788  df-topon 22805  df-t0 23207  df-kq 23588

This theorem is referenced by:  kqt0  23640  t0kq  23712