Theorem kqsat 23939

Description: Any open set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 23925). (Contributed by Mario Carneiro, 25-Aug-2015.)

Hypothesis

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

Assertion

Ref	Expression
kqsat	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (◡𝐹 “ (𝐹 “ 𝑈)) = 𝑈)

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

Allowed substitution hints:   𝑈(𝑥, 𝑦)   𝐹(𝑥, 𝑦)

Proof of Theorem kqsat

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		kqval.2	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
2	1	kqffn 23933	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
3		elpreima 7057	. . . . . 6 ⊢ (𝐹 Fn 𝑋 → (𝑧 ∈ (◡𝐹 “ (𝐹 “ 𝑈)) ↔ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ (𝐹 “ 𝑈))))
4	2, 3	syl 18	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑧 ∈ (◡𝐹 “ (𝐹 “ 𝑈)) ↔ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ (𝐹 “ 𝑈))))
5	4	adantr 486	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (𝑧 ∈ (◡𝐹 “ (𝐹 “ 𝑈)) ↔ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ (𝐹 “ 𝑈))))
6	1	kqfvima 23938	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝑧 ∈ 𝑋) → (𝑧 ∈ 𝑈 ↔ (𝐹‘𝑧) ∈ (𝐹 “ 𝑈)))
7	6	3expa 1136	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) ∧ 𝑧 ∈ 𝑋) → (𝑧 ∈ 𝑈 ↔ (𝐹‘𝑧) ∈ (𝐹 “ 𝑈)))
8	7	biimprd 251	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) ∈ (𝐹 “ 𝑈) → 𝑧 ∈ 𝑈))
9	8	expimpd 459	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → ((𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ (𝐹 “ 𝑈)) → 𝑧 ∈ 𝑈))
10	5, 9	sylbid 243	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (𝑧 ∈ (◡𝐹 “ (𝐹 “ 𝑈)) → 𝑧 ∈ 𝑈))
11	10	ssrdv 3944	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (◡𝐹 “ (𝐹 “ 𝑈)) ⊆ 𝑈)
12		toponss 23134	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → 𝑈 ⊆ 𝑋)
13	2	fndmd 6644	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → dom 𝐹 = 𝑋)
14	13	adantr 486	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → dom 𝐹 = 𝑋)
15	12, 14	sseqtrrd 3975	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → 𝑈 ⊆ dom 𝐹)
16		sseqin2 4176	. . . 4 ⊢ (𝑈 ⊆ dom 𝐹 ↔ (dom 𝐹 ∩ 𝑈) = 𝑈)
17	15, 16	sylib 221	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (dom 𝐹 ∩ 𝑈) = 𝑈)
18		dminss 6152	. . 3 ⊢ (dom 𝐹 ∩ 𝑈) ⊆ (◡𝐹 “ (𝐹 “ 𝑈))
19	17, 18	eqsstrrdi 3983	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → 𝑈 ⊆ (◡𝐹 “ (𝐹 “ 𝑈)))
20	11, 19	eqssd 3955	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (◡𝐹 “ (𝐹 “ 𝑈)) = 𝑈)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  {crab 3418   ∩ cin 3905   ⊆ wss 3906   ↦ cmpt 5194  ^◡ccnv 5662  dom cdm 5663   “ cima 5666   Fn wfn 6535  ‘cfv 6540  TopOnctopon 23117

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-topon 23118

This theorem is used by:  kqopn  23942  kqreglem2  23950  kqnrmlem2  23952