Theorem kqt0lem 23744

Description: Lemma for kqt0 23754. (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 23742	. . . . . . . . 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 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 345	. . . . . . . . 9 ⊢ (𝑧 = (𝐹 “ 𝑤) → (((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧) ↔ ((𝐹‘𝑎) ∈ (𝐹 “ 𝑤) ↔ (𝐹‘𝑏) ∈ (𝐹 “ 𝑤))))
7	6	rspcv 3618	. . . . . . . 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 23738	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽 ∧ 𝑎 ∈ 𝑋) → (𝑎 ∈ 𝑤 ↔ (𝐹‘𝑎) ∈ (𝐹 “ 𝑤)))
10	9	3expa 1119	. . . . . . . . . 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 23738	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽 ∧ 𝑏 ∈ 𝑋) → (𝑏 ∈ 𝑤 ↔ (𝐹‘𝑏) ∈ (𝐹 “ 𝑤)))
13	12	3expa 1119	. . . . . . . . . 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 3154	. . . . 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 23732	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ ∀𝑦 ∈ 𝐽 (𝑎 ∈ 𝑦 ↔ 𝑏 ∈ 𝑦)))
20	19	3expb 1121	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ ∀𝑦 ∈ 𝐽 (𝑎 ∈ 𝑦 ↔ 𝑏 ∈ 𝑦)))
21		elequ2 2123	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (𝑎 ∈ 𝑦 ↔ 𝑎 ∈ 𝑤))
22		elequ2 2123	. . . . . . . 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 3237	. . . . . 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 3202	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧) → (𝐹‘𝑎) = (𝐹‘𝑏)))
28	1	kqffn 23733	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
29		eleq1 2829	. . . . . . . . . 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 3178	. . . . . . . 8 ⊢ (𝑢 = (𝐹‘𝑎) → (∀𝑧 ∈ (KQ‘𝐽)(𝑢 ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) ↔ ∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧)))
32		eqeq1 2741	. . . . . . . 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 3178	. . . . . 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 7108	. . . . 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 342	. . . . . . . . 9 ⊢ (𝑣 = (𝐹‘𝑏) → (((𝐹‘𝑎) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) ↔ ((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧)))
38	37	ralbidv 3178	. . . . . . . 8 ⊢ (𝑣 = (𝐹‘𝑏) → (∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) ↔ ∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧)))
39		eqeq2 2749	. . . . . . . 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 7108	. . . . . 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 3178	. . . . 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 23735	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
47		ist0-2 23352	. . 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 2108  ∀wral 3061  {crab 3436   ↦ cmpt 5225  ran crn 5686   “ cima 5688   Fn wfn 6556  ‘cfv 6561  TopOnctopon 22916  Kol2ct0 23314  KQckq 23701

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 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-qtop 17552  df-top 22900  df-topon 22917  df-t0 23321  df-kq 23702

This theorem is referenced by:  kqt0  23754  t0kq  23826