Theorem kqt0lem 23963

Description: Lemma for kqt0 23973. (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 23961	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽) → (𝐹 “ 𝑤) ∈ (KQ‘𝐽))
3	2	adantlr 728	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ 𝑤 ∈ 𝐽) → (𝐹 “ 𝑤) ∈ (KQ‘𝐽))
4		eleq2 2851	. . . . . . . . . 10 ⊢ (𝑧 = (𝐹 “ 𝑤) → ((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑎) ∈ (𝐹 “ 𝑤)))
5		eleq2 2851	. . . . . . . . . 10 ⊢ (𝑧 = (𝐹 “ 𝑤) → ((𝐹‘𝑏) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ (𝐹 “ 𝑤)))
6	4, 5	bibi12d 348	. . . . . . . . 9 ⊢ (𝑧 = (𝐹 “ 𝑤) → (((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧) ↔ ((𝐹‘𝑎) ∈ (𝐹 “ 𝑤) ↔ (𝐹‘𝑏) ∈ (𝐹 “ 𝑤))))
7	6	rspcv 3575	. . . . . . . 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 18	. . . . . . 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 23957	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽 ∧ 𝑎 ∈ 𝑋) → (𝑎 ∈ 𝑤 ↔ (𝐹‘𝑎) ∈ (𝐹 “ 𝑤)))
10	9	3expa 1136	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽) ∧ 𝑎 ∈ 𝑋) → (𝑎 ∈ 𝑤 ↔ (𝐹‘𝑎) ∈ (𝐹 “ 𝑤)))
11	10	adantrr 730	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎 ∈ 𝑤 ↔ (𝐹‘𝑎) ∈ (𝐹 “ 𝑤)))
12	1	kqfvima 23957	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽 ∧ 𝑏 ∈ 𝑋) → (𝑏 ∈ 𝑤 ↔ (𝐹‘𝑏) ∈ (𝐹 “ 𝑤)))
13	12	3expa 1136	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽) ∧ 𝑏 ∈ 𝑋) → (𝑏 ∈ 𝑤 ↔ (𝐹‘𝑏) ∈ (𝐹 “ 𝑤)))
14	13	adantrl 729	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑏 ∈ 𝑤 ↔ (𝐹‘𝑏) ∈ (𝐹 “ 𝑤)))
15	11, 14	bibi12d 348	. . . . . . . 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 665	. . . . . . 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 262	. . . . . 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 3164	. . . . 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 23951	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ ∀𝑦 ∈ 𝐽 (𝑎 ∈ 𝑦 ↔ 𝑏 ∈ 𝑦)))
20	19	3expb 1138	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ ∀𝑦 ∈ 𝐽 (𝑎 ∈ 𝑦 ↔ 𝑏 ∈ 𝑦)))
21		elequ2 2160	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (𝑎 ∈ 𝑦 ↔ 𝑎 ∈ 𝑤))
22		elequ2 2160	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (𝑏 ∈ 𝑦 ↔ 𝑏 ∈ 𝑤))
23	21, 22	bibi12d 348	. . . . . . 7 ⊢ (𝑦 = 𝑤 → ((𝑎 ∈ 𝑦 ↔ 𝑏 ∈ 𝑦) ↔ (𝑎 ∈ 𝑤 ↔ 𝑏 ∈ 𝑤)))
24	23	cbvralvw 3242	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐽 (𝑎 ∈ 𝑦 ↔ 𝑏 ∈ 𝑦) ↔ ∀𝑤 ∈ 𝐽 (𝑎 ∈ 𝑤 ↔ 𝑏 ∈ 𝑤))
25	20, 24	bitrdi 290	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ ∀𝑤 ∈ 𝐽 (𝑎 ∈ 𝑤 ↔ 𝑏 ∈ 𝑤)))
26	18, 25	sylibrd 262	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧) → (𝐹‘𝑎) = (𝐹‘𝑏)))
27	26	ralrimivva 3207	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧) → (𝐹‘𝑎) = (𝐹‘𝑏)))
28	1	kqffn 23952	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
29		eleq1 2850	. . . . . . . . . 10 ⊢ (𝑢 = (𝐹‘𝑎) → (𝑢 ∈ 𝑧 ↔ (𝐹‘𝑎) ∈ 𝑧))
30	29	bibi1d 346	. . . . . . . . 9 ⊢ (𝑢 = (𝐹‘𝑎) → ((𝑢 ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) ↔ ((𝐹‘𝑎) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧)))
31	30	ralbidv 3187	. . . . . . . 8 ⊢ (𝑢 = (𝐹‘𝑎) → (∀𝑧 ∈ (KQ‘𝐽)(𝑢 ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) ↔ ∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧)))
32		eqeq1 2766	. . . . . . . 8 ⊢ (𝑢 = (𝐹‘𝑎) → (𝑢 = 𝑣 ↔ (𝐹‘𝑎) = 𝑣))
33	31, 32	imbi12d 347	. . . . . . 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 3187	. . . . . 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 7084	. . . . 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 2850	. . . . . . . . . 10 ⊢ (𝑣 = (𝐹‘𝑏) → (𝑣 ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧))
37	36	bibi2d 345	. . . . . . . . 9 ⊢ (𝑣 = (𝐹‘𝑏) → (((𝐹‘𝑎) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) ↔ ((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧)))
38	37	ralbidv 3187	. . . . . . . 8 ⊢ (𝑣 = (𝐹‘𝑏) → (∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) ↔ ∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧)))
39		eqeq2 2774	. . . . . . . 8 ⊢ (𝑣 = (𝐹‘𝑏) → ((𝐹‘𝑎) = 𝑣 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)))
40	38, 39	imbi12d 347	. . . . . . 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 7084	. . . . . 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 3187	. . . . 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 282	. . . 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 18	. . 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 260	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∀𝑢 ∈ ran 𝐹∀𝑣 ∈ ran 𝐹(∀𝑧 ∈ (KQ‘𝐽)(𝑢 ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) → 𝑢 = 𝑣))
46	1	kqtopon 23954	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
47		ist0-2 23570	. . 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 18	. 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 260	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ Kol2)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  {crab 3414   ↦ cmpt 5190  ran crn 5660   “ cima 5662   Fn wfn 6532  ‘cfv 6537  TopOnctopon 23136  Kol2ct0 23532  KQckq 23920

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-qtop 17597  df-top 23120  df-topon 23137  df-t0 23539  df-kq 23921

This theorem is used by:  kqt0  23973  t0kq  24045