Theorem kqopn 22342

Description: The topological indistinguishability map is an open map. (Contributed by Mario Carneiro, 25-Aug-2015.)

Hypothesis

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

Assertion

Ref	Expression
kqopn	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (𝐹 “ 𝑈) ∈ (KQ‘𝐽))

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

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

Proof of Theorem kqopn

Step	Hyp	Ref	Expression
1		imassrn 5940	. . . 4 ⊢ (𝐹 “ 𝑈) ⊆ ran 𝐹
2	1	a1i 11	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (𝐹 “ 𝑈) ⊆ ran 𝐹)
3		kqval.2	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
4	3	kqsat 22339	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (◡𝐹 “ (𝐹 “ 𝑈)) = 𝑈)
5		simpr 487	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → 𝑈 ∈ 𝐽)
6	4, 5	eqeltrd 2913	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (◡𝐹 “ (𝐹 “ 𝑈)) ∈ 𝐽)
7	3	kqffn 22333	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
8		dffn4 6596	. . . . . 6 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–onto→ran 𝐹)
9	7, 8	sylib 220	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹:𝑋–onto→ran 𝐹)
10	9	adantr 483	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → 𝐹:𝑋–onto→ran 𝐹)
11		elqtop3 22311	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→ran 𝐹) → ((𝐹 “ 𝑈) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹 “ 𝑈) ⊆ ran 𝐹 ∧ (◡𝐹 “ (𝐹 “ 𝑈)) ∈ 𝐽)))
12	10, 11	syldan 593	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → ((𝐹 “ 𝑈) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹 “ 𝑈) ⊆ ran 𝐹 ∧ (◡𝐹 “ (𝐹 “ 𝑈)) ∈ 𝐽)))
13	2, 6, 12	mpbir2and 711	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (𝐹 “ 𝑈) ∈ (𝐽 qTop 𝐹))
14	3	kqval 22334	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
15	14	adantr 483	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
16	13, 15	eleqtrrd 2916	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (𝐹 “ 𝑈) ∈ (KQ‘𝐽))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  {crab 3142   ⊆ wss 3936   ↦ cmpt 5146  ^◡ccnv 5554  ran crn 5556   “ cima 5558   Fn wfn 6350  –onto→wfo 6353  ‘cfv 6355  (class class class)co 7156   qTop cqtop 16776  TopOnctopon 21518  KQckq 22301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-qtop 16780  df-topon 21519  df-kq 22302

This theorem is referenced by:  kqt0lem  22344  isr0  22345  regr1lem  22347  kqreglem1  22349  kqreglem2  22350  kqnrmlem1  22351  kqnrmlem2  22352