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Theorem qtoptop2 22235
Description: The quotient topology is a topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
Assertion
Ref Expression
qtoptop2 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝐽 qTop 𝐹) ∈ Top)

Proof of Theorem qtoptop2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2818 . . . 4 𝐽 = 𝐽
21qtopres 22234 . . 3 (𝐹𝑉 → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹 𝐽)))
323ad2ant2 1126 . 2 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹 𝐽)))
4 simp1 1128 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → 𝐽 ∈ Top)
5 funres 6390 . . . . . . . . . . . . . . 15 (Fun 𝐹 → Fun (𝐹 𝐽))
653ad2ant3 1127 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → Fun (𝐹 𝐽))
7 funforn 6590 . . . . . . . . . . . . . 14 (Fun (𝐹 𝐽) ↔ (𝐹 𝐽):dom (𝐹 𝐽)–onto→ran (𝐹 𝐽))
86, 7sylib 219 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝐹 𝐽):dom (𝐹 𝐽)–onto→ran (𝐹 𝐽))
9 dmres 5868 . . . . . . . . . . . . . . 15 dom (𝐹 𝐽) = ( 𝐽 ∩ dom 𝐹)
10 inss1 4202 . . . . . . . . . . . . . . 15 ( 𝐽 ∩ dom 𝐹) ⊆ 𝐽
119, 10eqsstri 3998 . . . . . . . . . . . . . 14 dom (𝐹 𝐽) ⊆ 𝐽
1211a1i 11 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → dom (𝐹 𝐽) ⊆ 𝐽)
131elqtop 22233 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ (𝐹 𝐽):dom (𝐹 𝐽)–onto→ran (𝐹 𝐽) ∧ dom (𝐹 𝐽) ⊆ 𝐽) → (𝑦 ∈ (𝐽 qTop (𝐹 𝐽)) ↔ (𝑦 ⊆ ran (𝐹 𝐽) ∧ ((𝐹 𝐽) “ 𝑦) ∈ 𝐽)))
144, 8, 12, 13syl3anc 1363 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝑦 ∈ (𝐽 qTop (𝐹 𝐽)) ↔ (𝑦 ⊆ ran (𝐹 𝐽) ∧ ((𝐹 𝐽) “ 𝑦) ∈ 𝐽)))
1514simprbda 499 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽))) → 𝑦 ⊆ ran (𝐹 𝐽))
16 velpw 4543 . . . . . . . . . . 11 (𝑦 ∈ 𝒫 ran (𝐹 𝐽) ↔ 𝑦 ⊆ ran (𝐹 𝐽))
1715, 16sylibr 235 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽))) → 𝑦 ∈ 𝒫 ran (𝐹 𝐽))
1817ex 413 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝑦 ∈ (𝐽 qTop (𝐹 𝐽)) → 𝑦 ∈ 𝒫 ran (𝐹 𝐽)))
1918ssrdv 3970 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝐽 qTop (𝐹 𝐽)) ⊆ 𝒫 ran (𝐹 𝐽))
20 sstr2 3971 . . . . . . . 8 (𝑥 ⊆ (𝐽 qTop (𝐹 𝐽)) → ((𝐽 qTop (𝐹 𝐽)) ⊆ 𝒫 ran (𝐹 𝐽) → 𝑥 ⊆ 𝒫 ran (𝐹 𝐽)))
2119, 20syl5com 31 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝑥 ⊆ (𝐽 qTop (𝐹 𝐽)) → 𝑥 ⊆ 𝒫 ran (𝐹 𝐽)))
22 sspwuni 5013 . . . . . . 7 (𝑥 ⊆ 𝒫 ran (𝐹 𝐽) ↔ 𝑥 ⊆ ran (𝐹 𝐽))
2321, 22syl6ib 252 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝑥 ⊆ (𝐽 qTop (𝐹 𝐽)) → 𝑥 ⊆ ran (𝐹 𝐽)))
24 imauni 6996 . . . . . . . 8 ((𝐹 𝐽) “ 𝑥) = 𝑦𝑥 ((𝐹 𝐽) “ 𝑦)
2514simplbda 500 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽))) → ((𝐹 𝐽) “ 𝑦) ∈ 𝐽)
2625ralrimiva 3179 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → ∀𝑦 ∈ (𝐽 qTop (𝐹 𝐽))((𝐹 𝐽) “ 𝑦) ∈ 𝐽)
27 ssralv 4030 . . . . . . . . . 10 (𝑥 ⊆ (𝐽 qTop (𝐹 𝐽)) → (∀𝑦 ∈ (𝐽 qTop (𝐹 𝐽))((𝐹 𝐽) “ 𝑦) ∈ 𝐽 → ∀𝑦𝑥 ((𝐹 𝐽) “ 𝑦) ∈ 𝐽))
2826, 27mpan9 507 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ 𝑥 ⊆ (𝐽 qTop (𝐹 𝐽))) → ∀𝑦𝑥 ((𝐹 𝐽) “ 𝑦) ∈ 𝐽)
29 iunopn 21434 . . . . . . . . 9 ((𝐽 ∈ Top ∧ ∀𝑦𝑥 ((𝐹 𝐽) “ 𝑦) ∈ 𝐽) → 𝑦𝑥 ((𝐹 𝐽) “ 𝑦) ∈ 𝐽)
304, 28, 29syl2an2r 681 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ 𝑥 ⊆ (𝐽 qTop (𝐹 𝐽))) → 𝑦𝑥 ((𝐹 𝐽) “ 𝑦) ∈ 𝐽)
3124, 30eqeltrid 2914 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ 𝑥 ⊆ (𝐽 qTop (𝐹 𝐽))) → ((𝐹 𝐽) “ 𝑥) ∈ 𝐽)
3231ex 413 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝑥 ⊆ (𝐽 qTop (𝐹 𝐽)) → ((𝐹 𝐽) “ 𝑥) ∈ 𝐽))
3323, 32jcad 513 . . . . 5 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝑥 ⊆ (𝐽 qTop (𝐹 𝐽)) → ( 𝑥 ⊆ ran (𝐹 𝐽) ∧ ((𝐹 𝐽) “ 𝑥) ∈ 𝐽)))
341elqtop 22233 . . . . . 6 ((𝐽 ∈ Top ∧ (𝐹 𝐽):dom (𝐹 𝐽)–onto→ran (𝐹 𝐽) ∧ dom (𝐹 𝐽) ⊆ 𝐽) → ( 𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ↔ ( 𝑥 ⊆ ran (𝐹 𝐽) ∧ ((𝐹 𝐽) “ 𝑥) ∈ 𝐽)))
354, 8, 12, 34syl3anc 1363 . . . . 5 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → ( 𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ↔ ( 𝑥 ⊆ ran (𝐹 𝐽) ∧ ((𝐹 𝐽) “ 𝑥) ∈ 𝐽)))
3633, 35sylibrd 260 . . . 4 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝑥 ⊆ (𝐽 qTop (𝐹 𝐽)) → 𝑥 ∈ (𝐽 qTop (𝐹 𝐽))))
3736alrimiv 1919 . . 3 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → ∀𝑥(𝑥 ⊆ (𝐽 qTop (𝐹 𝐽)) → 𝑥 ∈ (𝐽 qTop (𝐹 𝐽))))
38 inss1 4202 . . . . . 6 (𝑥𝑦) ⊆ 𝑥
391elqtop 22233 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝐹 𝐽):dom (𝐹 𝐽)–onto→ran (𝐹 𝐽) ∧ dom (𝐹 𝐽) ⊆ 𝐽) → (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ↔ (𝑥 ⊆ ran (𝐹 𝐽) ∧ ((𝐹 𝐽) “ 𝑥) ∈ 𝐽)))
404, 8, 12, 39syl3anc 1363 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ↔ (𝑥 ⊆ ran (𝐹 𝐽) ∧ ((𝐹 𝐽) “ 𝑥) ∈ 𝐽)))
4140biimpa 477 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ 𝑥 ∈ (𝐽 qTop (𝐹 𝐽))) → (𝑥 ⊆ ran (𝐹 𝐽) ∧ ((𝐹 𝐽) “ 𝑥) ∈ 𝐽))
4241adantrr 713 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → (𝑥 ⊆ ran (𝐹 𝐽) ∧ ((𝐹 𝐽) “ 𝑥) ∈ 𝐽))
4342simpld 495 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → 𝑥 ⊆ ran (𝐹 𝐽))
4438, 43sstrid 3975 . . . . 5 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → (𝑥𝑦) ⊆ ran (𝐹 𝐽))
456adantr 481 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → Fun (𝐹 𝐽))
46 inpreima 6826 . . . . . . 7 (Fun (𝐹 𝐽) → ((𝐹 𝐽) “ (𝑥𝑦)) = (((𝐹 𝐽) “ 𝑥) ∩ ((𝐹 𝐽) “ 𝑦)))
4745, 46syl 17 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → ((𝐹 𝐽) “ (𝑥𝑦)) = (((𝐹 𝐽) “ 𝑥) ∩ ((𝐹 𝐽) “ 𝑦)))
484adantr 481 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → 𝐽 ∈ Top)
4942simprd 496 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → ((𝐹 𝐽) “ 𝑥) ∈ 𝐽)
5025adantrl 712 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → ((𝐹 𝐽) “ 𝑦) ∈ 𝐽)
51 inopn 21435 . . . . . . 7 ((𝐽 ∈ Top ∧ ((𝐹 𝐽) “ 𝑥) ∈ 𝐽 ∧ ((𝐹 𝐽) “ 𝑦) ∈ 𝐽) → (((𝐹 𝐽) “ 𝑥) ∩ ((𝐹 𝐽) “ 𝑦)) ∈ 𝐽)
5248, 49, 50, 51syl3anc 1363 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → (((𝐹 𝐽) “ 𝑥) ∩ ((𝐹 𝐽) “ 𝑦)) ∈ 𝐽)
5347, 52eqeltrd 2910 . . . . 5 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → ((𝐹 𝐽) “ (𝑥𝑦)) ∈ 𝐽)
541elqtop 22233 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝐹 𝐽):dom (𝐹 𝐽)–onto→ran (𝐹 𝐽) ∧ dom (𝐹 𝐽) ⊆ 𝐽) → ((𝑥𝑦) ∈ (𝐽 qTop (𝐹 𝐽)) ↔ ((𝑥𝑦) ⊆ ran (𝐹 𝐽) ∧ ((𝐹 𝐽) “ (𝑥𝑦)) ∈ 𝐽)))
554, 8, 12, 54syl3anc 1363 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → ((𝑥𝑦) ∈ (𝐽 qTop (𝐹 𝐽)) ↔ ((𝑥𝑦) ⊆ ran (𝐹 𝐽) ∧ ((𝐹 𝐽) “ (𝑥𝑦)) ∈ 𝐽)))
5655adantr 481 . . . . 5 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → ((𝑥𝑦) ∈ (𝐽 qTop (𝐹 𝐽)) ↔ ((𝑥𝑦) ⊆ ran (𝐹 𝐽) ∧ ((𝐹 𝐽) “ (𝑥𝑦)) ∈ 𝐽)))
5744, 53, 56mpbir2and 709 . . . 4 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → (𝑥𝑦) ∈ (𝐽 qTop (𝐹 𝐽)))
5857ralrimivva 3188 . . 3 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → ∀𝑥 ∈ (𝐽 qTop (𝐹 𝐽))∀𝑦 ∈ (𝐽 qTop (𝐹 𝐽))(𝑥𝑦) ∈ (𝐽 qTop (𝐹 𝐽)))
59 ovex 7178 . . . 4 (𝐽 qTop (𝐹 𝐽)) ∈ V
60 istopg 21431 . . . 4 ((𝐽 qTop (𝐹 𝐽)) ∈ V → ((𝐽 qTop (𝐹 𝐽)) ∈ Top ↔ (∀𝑥(𝑥 ⊆ (𝐽 qTop (𝐹 𝐽)) → 𝑥 ∈ (𝐽 qTop (𝐹 𝐽))) ∧ ∀𝑥 ∈ (𝐽 qTop (𝐹 𝐽))∀𝑦 ∈ (𝐽 qTop (𝐹 𝐽))(𝑥𝑦) ∈ (𝐽 qTop (𝐹 𝐽)))))
6159, 60ax-mp 5 . . 3 ((𝐽 qTop (𝐹 𝐽)) ∈ Top ↔ (∀𝑥(𝑥 ⊆ (𝐽 qTop (𝐹 𝐽)) → 𝑥 ∈ (𝐽 qTop (𝐹 𝐽))) ∧ ∀𝑥 ∈ (𝐽 qTop (𝐹 𝐽))∀𝑦 ∈ (𝐽 qTop (𝐹 𝐽))(𝑥𝑦) ∈ (𝐽 qTop (𝐹 𝐽))))
6237, 58, 61sylanbrc 583 . 2 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝐽 qTop (𝐹 𝐽)) ∈ Top)
633, 62eqeltrd 2910 1 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝐽 qTop 𝐹) ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079  wal 1526   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  cin 3932  wss 3933  𝒫 cpw 4535   cuni 4830   ciun 4910  ccnv 5547  dom cdm 5548  ran crn 5549  cres 5550  cima 5551  Fun wfun 6342  ontowfo 6346  (class class class)co 7145   qTop cqtop 16764  Topctop 21429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-qtop 16768  df-top 21430
This theorem is referenced by:  qtoptop  22236
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