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Theorem qtoptop2 23050
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 2736 . . . 4 𝐽 = 𝐽
21qtopres 23049 . . 3 (𝐹𝑉 → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹 𝐽)))
323ad2ant2 1134 . 2 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹 𝐽)))
4 simp1 1136 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → 𝐽 ∈ Top)
5 funres 6543 . . . . . . . . . . . . . . 15 (Fun 𝐹 → Fun (𝐹 𝐽))
653ad2ant3 1135 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → Fun (𝐹 𝐽))
7 funforn 6763 . . . . . . . . . . . . . 14 (Fun (𝐹 𝐽) ↔ (𝐹 𝐽):dom (𝐹 𝐽)–onto→ran (𝐹 𝐽))
86, 7sylib 217 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝐹 𝐽):dom (𝐹 𝐽)–onto→ran (𝐹 𝐽))
9 dmres 5959 . . . . . . . . . . . . . . 15 dom (𝐹 𝐽) = ( 𝐽 ∩ dom 𝐹)
10 inss1 4188 . . . . . . . . . . . . . . 15 ( 𝐽 ∩ dom 𝐹) ⊆ 𝐽
119, 10eqsstri 3978 . . . . . . . . . . . . . 14 dom (𝐹 𝐽) ⊆ 𝐽
1211a1i 11 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → dom (𝐹 𝐽) ⊆ 𝐽)
131elqtop 23048 . . . . . . . . . . . . 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 1371 . . . . . . . . . . . 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 4565 . . . . . . . . . . 11 (𝑦 ∈ 𝒫 ran (𝐹 𝐽) ↔ 𝑦 ⊆ ran (𝐹 𝐽))
1715, 16sylibr 233 . . . . . . . . . 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 3950 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝐽 qTop (𝐹 𝐽)) ⊆ 𝒫 ran (𝐹 𝐽))
20 sstr2 3951 . . . . . . . 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 5060 . . . . . . 7 (𝑥 ⊆ 𝒫 ran (𝐹 𝐽) ↔ 𝑥 ⊆ ran (𝐹 𝐽))
2321, 22syl6ib 250 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝑥 ⊆ (𝐽 qTop (𝐹 𝐽)) → 𝑥 ⊆ ran (𝐹 𝐽)))
24 imauni 7193 . . . . . . . 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 3143 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → ∀𝑦 ∈ (𝐽 qTop (𝐹 𝐽))((𝐹 𝐽) “ 𝑦) ∈ 𝐽)
27 ssralv 4010 . . . . . . . . . 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 22247 . . . . . . . . 9 ((𝐽 ∈ Top ∧ ∀𝑦𝑥 ((𝐹 𝐽) “ 𝑦) ∈ 𝐽) → 𝑦𝑥 ((𝐹 𝐽) “ 𝑦) ∈ 𝐽)
304, 28, 29syl2an2r 683 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ 𝑥 ⊆ (𝐽 qTop (𝐹 𝐽))) → 𝑦𝑥 ((𝐹 𝐽) “ 𝑦) ∈ 𝐽)
3124, 30eqeltrid 2842 . . . . . . 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 23048 . . . . . 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 1371 . . . . 5 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → ( 𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ↔ ( 𝑥 ⊆ ran (𝐹 𝐽) ∧ ((𝐹 𝐽) “ 𝑥) ∈ 𝐽)))
3633, 35sylibrd 258 . . . 4 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝑥 ⊆ (𝐽 qTop (𝐹 𝐽)) → 𝑥 ∈ (𝐽 qTop (𝐹 𝐽))))
3736alrimiv 1930 . . 3 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → ∀𝑥(𝑥 ⊆ (𝐽 qTop (𝐹 𝐽)) → 𝑥 ∈ (𝐽 qTop (𝐹 𝐽))))
38 inss1 4188 . . . . . 6 (𝑥𝑦) ⊆ 𝑥
391elqtop 23048 . . . . . . . . . 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 1371 . . . . . . . . 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 715 . . . . . . 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 3955 . . . . 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 7014 . . . . . . 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 714 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → ((𝐹 𝐽) “ 𝑦) ∈ 𝐽)
51 inopn 22248 . . . . . . 7 ((𝐽 ∈ Top ∧ ((𝐹 𝐽) “ 𝑥) ∈ 𝐽 ∧ ((𝐹 𝐽) “ 𝑦) ∈ 𝐽) → (((𝐹 𝐽) “ 𝑥) ∩ ((𝐹 𝐽) “ 𝑦)) ∈ 𝐽)
5248, 49, 50, 51syl3anc 1371 . . . . . 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 2838 . . . . 5 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → ((𝐹 𝐽) “ (𝑥𝑦)) ∈ 𝐽)
541elqtop 23048 . . . . . . 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 1371 . . . . . 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 711 . . . 4 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → (𝑥𝑦) ∈ (𝐽 qTop (𝐹 𝐽)))
5857ralrimivva 3197 . . 3 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → ∀𝑥 ∈ (𝐽 qTop (𝐹 𝐽))∀𝑦 ∈ (𝐽 qTop (𝐹 𝐽))(𝑥𝑦) ∈ (𝐽 qTop (𝐹 𝐽)))
59 ovex 7390 . . . 4 (𝐽 qTop (𝐹 𝐽)) ∈ V
60 istopg 22244 . . . 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 2838 1 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝐽 qTop 𝐹) ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087  wal 1539   = wceq 1541  wcel 2106  wral 3064  Vcvv 3445  cin 3909  wss 3910  𝒫 cpw 4560   cuni 4865   ciun 4954  ccnv 5632  dom cdm 5633  ran crn 5634  cres 5635  cima 5636  Fun wfun 6490  ontowfo 6494  (class class class)co 7357   qTop cqtop 17385  Topctop 22242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-qtop 17389  df-top 22243
This theorem is referenced by:  qtoptop  23051
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