Theorem kqt0lem 21760

Description: Lemma for kqt0 21770. (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 21758	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽) → (𝐹 “ 𝑤) ∈ (KQ‘𝐽))
3	2	adantlr 694	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ 𝑤 ∈ 𝐽) → (𝐹 “ 𝑤) ∈ (KQ‘𝐽))
4		eleq2 2839	. . . . . . . . . 10 ⊢ (𝑧 = (𝐹 “ 𝑤) → ((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑎) ∈ (𝐹 “ 𝑤)))
5		eleq2 2839	. . . . . . . . . 10 ⊢ (𝑧 = (𝐹 “ 𝑤) → ((𝐹‘𝑏) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ (𝐹 “ 𝑤)))
6	4, 5	bibi12d 334	. . . . . . . . 9 ⊢ (𝑧 = (𝐹 “ 𝑤) → (((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧) ↔ ((𝐹‘𝑎) ∈ (𝐹 “ 𝑤) ↔ (𝐹‘𝑏) ∈ (𝐹 “ 𝑤))))
7	6	rspcv 3456	. . . . . . . 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 21754	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽 ∧ 𝑎 ∈ 𝑋) → (𝑎 ∈ 𝑤 ↔ (𝐹‘𝑎) ∈ (𝐹 “ 𝑤)))
10	9	3expa 1111	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽) ∧ 𝑎 ∈ 𝑋) → (𝑎 ∈ 𝑤 ↔ (𝐹‘𝑎) ∈ (𝐹 “ 𝑤)))
11	10	adantrr 696	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎 ∈ 𝑤 ↔ (𝐹‘𝑎) ∈ (𝐹 “ 𝑤)))
12	1	kqfvima 21754	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽 ∧ 𝑏 ∈ 𝑋) → (𝑏 ∈ 𝑤 ↔ (𝐹‘𝑏) ∈ (𝐹 “ 𝑤)))
13	12	3expa 1111	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽) ∧ 𝑏 ∈ 𝑋) → (𝑏 ∈ 𝑤 ↔ (𝐹‘𝑏) ∈ (𝐹 “ 𝑤)))
14	13	adantrl 695	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑏 ∈ 𝑤 ↔ (𝐹‘𝑏) ∈ (𝐹 “ 𝑤)))
15	11, 14	bibi12d 334	. . . . . . . 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 631	. . . . . . 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 249	. . . . . 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 3118	. . . . 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 21748	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ ∀𝑦 ∈ 𝐽 (𝑎 ∈ 𝑦 ↔ 𝑏 ∈ 𝑦)))
20	19	3expb 1113	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ ∀𝑦 ∈ 𝐽 (𝑎 ∈ 𝑦 ↔ 𝑏 ∈ 𝑦)))
21		elequ2 2159	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (𝑎 ∈ 𝑦 ↔ 𝑎 ∈ 𝑤))
22		elequ2 2159	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (𝑏 ∈ 𝑦 ↔ 𝑏 ∈ 𝑤))
23	21, 22	bibi12d 334	. . . . . . 7 ⊢ (𝑦 = 𝑤 → ((𝑎 ∈ 𝑦 ↔ 𝑏 ∈ 𝑦) ↔ (𝑎 ∈ 𝑤 ↔ 𝑏 ∈ 𝑤)))
24	23	cbvralv 3320	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐽 (𝑎 ∈ 𝑦 ↔ 𝑏 ∈ 𝑦) ↔ ∀𝑤 ∈ 𝐽 (𝑎 ∈ 𝑤 ↔ 𝑏 ∈ 𝑤))
25	20, 24	syl6bb 276	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ ∀𝑤 ∈ 𝐽 (𝑎 ∈ 𝑤 ↔ 𝑏 ∈ 𝑤)))
26	18, 25	sylibrd 249	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧) → (𝐹‘𝑎) = (𝐹‘𝑏)))
27	26	ralrimivva 3120	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧) → (𝐹‘𝑎) = (𝐹‘𝑏)))
28	1	kqffn 21749	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
29		eleq1 2838	. . . . . . . . . 10 ⊢ (𝑢 = (𝐹‘𝑎) → (𝑢 ∈ 𝑧 ↔ (𝐹‘𝑎) ∈ 𝑧))
30	29	bibi1d 332	. . . . . . . . 9 ⊢ (𝑢 = (𝐹‘𝑎) → ((𝑢 ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) ↔ ((𝐹‘𝑎) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧)))
31	30	ralbidv 3135	. . . . . . . 8 ⊢ (𝑢 = (𝐹‘𝑎) → (∀𝑧 ∈ (KQ‘𝐽)(𝑢 ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) ↔ ∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧)))
32		eqeq1 2775	. . . . . . . 8 ⊢ (𝑢 = (𝐹‘𝑎) → (𝑢 = 𝑣 ↔ (𝐹‘𝑎) = 𝑣))
33	31, 32	imbi12d 333	. . . . . . 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 3135	. . . . . 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 6505	. . . . 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 2838	. . . . . . . . . 10 ⊢ (𝑣 = (𝐹‘𝑏) → (𝑣 ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧))
37	36	bibi2d 331	. . . . . . . . 9 ⊢ (𝑣 = (𝐹‘𝑏) → (((𝐹‘𝑎) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) ↔ ((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧)))
38	37	ralbidv 3135	. . . . . . . 8 ⊢ (𝑣 = (𝐹‘𝑏) → (∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) ↔ ∀𝑧 ∈ (KQ‘𝐽)((𝐹‘𝑎) ∈ 𝑧 ↔ (𝐹‘𝑏) ∈ 𝑧)))
39		eqeq2 2782	. . . . . . . 8 ⊢ (𝑣 = (𝐹‘𝑏) → ((𝐹‘𝑎) = 𝑣 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)))
40	38, 39	imbi12d 333	. . . . . . 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 6505	. . . . . 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 3135	. . . . 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 268	. . . 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 247	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∀𝑢 ∈ ran 𝐹∀𝑣 ∈ ran 𝐹(∀𝑧 ∈ (KQ‘𝐽)(𝑢 ∈ 𝑧 ↔ 𝑣 ∈ 𝑧) → 𝑢 = 𝑣))
46	1	kqtopon 21751	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
47		ist0-2 21369	. . 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 247	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ Kol2)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1631   ∈ wcel 2145  ∀wral 3061  {crab 3065   ↦ cmpt 4863  ran crn 5250   “ cima 5252   Fn wfn 6026  ‘cfv 6031  TopOnctopon 20935  Kol2ct0 21331  KQckq 21717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-qtop 16375  df-top 20919  df-topon 20936  df-t0 21338  df-kq 21718

This theorem is referenced by:  kqt0  21770  t0kq  21842