Theorem kqval 23852

Description: Value of the quotient topology function. (Contributed by Mario Carneiro, 25-Aug-2015.)

Hypothesis

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

Assertion

Ref	Expression
kqval	⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) = (𝐽 qTop 𝐹))

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

Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem kqval

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		topontop 23039	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2		id 23	. . . . 5 ⊢ (𝑗 = 𝐽 → 𝑗 = 𝐽)
3		unieq 4887	. . . . . 6 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
4		rabeq 3437	. . . . . 6 ⊢ (𝑗 = 𝐽 → {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦} = {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
5	3, 4	mpteq12dv 5202	. . . . 5 ⊢ (𝑗 = 𝐽 → (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦}) = (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}))
6	2, 5	oveq12d 7429	. . . 4 ⊢ (𝑗 = 𝐽 → (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦})) = (𝐽 qTop (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})))
7		df-kq 23820	. . . 4 ⊢ KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦})))
8		ovex 7444	. . . 4 ⊢ (𝐽 qTop (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})) ∈ V
9	6, 7, 8	fvmpt 6990	. . 3 ⊢ (𝐽 ∈ Top → (KQ‘𝐽) = (𝐽 qTop (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})))
10	1, 9	syl 18	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) = (𝐽 qTop (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})))
11		kqval.2	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
12		toponuni 23040	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
13	12	mpteq1d 5205	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) = (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}))
14	11, 13	eqtrid 2816	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 = (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}))
15	14	oveq2d 7427	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 qTop 𝐹) = (𝐽 qTop (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})))
16	10, 15	eqtr4d 2807	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) = (𝐽 qTop 𝐹))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  {crab 3423  ∪ cuni 4876   ↦ cmpt 5196  ‘cfv 6537  (class class class)co 7411   qTop cqtop 17557  Topctop 23019  TopOnctopon 23036  KQckq 23819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7414  df-topon 23037  df-kq 23820

This theorem is referenced by:  kqtopon  23853  kqid  23854  kqopn  23860  kqcld  23861  t0kq  23944