Theorem kqid 23712

Description: The topological indistinguishability map is a continuous function into the Kolmogorov quotient. (Contributed by Mario Carneiro, 25-Aug-2015.)

Hypothesis

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

Assertion

Ref	Expression
kqid	⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))

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

Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem kqid

Step	Hyp	Ref	Expression
1		kqval.2	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
2	1	kqffn 23709	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
3		qtopid 23689	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
4	2, 3	mpdan 693	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
5	1	kqval 23710	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
6	5	oveq2d 7373	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 Cn (KQ‘𝐽)) = (𝐽 Cn (𝐽 qTop 𝐹)))
7	4, 6	eleqtrrd 2842	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  {crab 3391   ↦ cmpt 5154   Fn wfn 6481  ‘cfv 6486  (class class class)co 7357   qTop cqtop 17459  TopOnctopon 22894   Cn ccn 23208  KQckq 23677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7360  df-oprab 7361  df-mpo 7362  df-map 8766  df-qtop 17463  df-top 22878  df-topon 22895  df-cn 23211  df-kq 23678

This theorem is referenced by:  isr0  23721  r0cld  23722  kqreglem1  23725  kqreglem2  23726  kqnrmlem1  23727  kqnrmlem2  23728