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Theorem qtoptop2 23707
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 2737 . . . 4 𝐽 = 𝐽
21qtopres 23706 . . 3 (𝐹𝑉 → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹 𝐽)))
323ad2ant2 1135 . 2 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹 𝐽)))
4 simp1 1137 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → 𝐽 ∈ Top)
5 funres 6608 . . . . . . . . . . . . . . 15 (Fun 𝐹 → Fun (𝐹 𝐽))
653ad2ant3 1136 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → Fun (𝐹 𝐽))
7 funforn 6827 . . . . . . . . . . . . . 14 (Fun (𝐹 𝐽) ↔ (𝐹 𝐽):dom (𝐹 𝐽)–onto→ran (𝐹 𝐽))
86, 7sylib 218 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝐹 𝐽):dom (𝐹 𝐽)–onto→ran (𝐹 𝐽))
9 dmres 6030 . . . . . . . . . . . . . . 15 dom (𝐹 𝐽) = ( 𝐽 ∩ dom 𝐹)
10 inss1 4237 . . . . . . . . . . . . . . 15 ( 𝐽 ∩ dom 𝐹) ⊆ 𝐽
119, 10eqsstri 4030 . . . . . . . . . . . . . 14 dom (𝐹 𝐽) ⊆ 𝐽
1211a1i 11 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → dom (𝐹 𝐽) ⊆ 𝐽)
131elqtop 23705 . . . . . . . . . . . . 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 1373 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝑦 ∈ (𝐽 qTop (𝐹 𝐽)) ↔ (𝑦 ⊆ ran (𝐹 𝐽) ∧ ((𝐹 𝐽) “ 𝑦) ∈ 𝐽)))
1514simprbda 498 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽))) → 𝑦 ⊆ ran (𝐹 𝐽))
16 velpw 4605 . . . . . . . . . . 11 (𝑦 ∈ 𝒫 ran (𝐹 𝐽) ↔ 𝑦 ⊆ ran (𝐹 𝐽))
1715, 16sylibr 234 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽))) → 𝑦 ∈ 𝒫 ran (𝐹 𝐽))
1817ex 412 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝑦 ∈ (𝐽 qTop (𝐹 𝐽)) → 𝑦 ∈ 𝒫 ran (𝐹 𝐽)))
1918ssrdv 3989 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝐽 qTop (𝐹 𝐽)) ⊆ 𝒫 ran (𝐹 𝐽))
20 sstr2 3990 . . . . . . . 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 5100 . . . . . . 7 (𝑥 ⊆ 𝒫 ran (𝐹 𝐽) ↔ 𝑥 ⊆ ran (𝐹 𝐽))
2321, 22imbitrdi 251 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝑥 ⊆ (𝐽 qTop (𝐹 𝐽)) → 𝑥 ⊆ ran (𝐹 𝐽)))
24 imauni 7266 . . . . . . . 8 ((𝐹 𝐽) “ 𝑥) = 𝑦𝑥 ((𝐹 𝐽) “ 𝑦)
2514simplbda 499 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽))) → ((𝐹 𝐽) “ 𝑦) ∈ 𝐽)
2625ralrimiva 3146 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → ∀𝑦 ∈ (𝐽 qTop (𝐹 𝐽))((𝐹 𝐽) “ 𝑦) ∈ 𝐽)
27 ssralv 4052 . . . . . . . . . 10 (𝑥 ⊆ (𝐽 qTop (𝐹 𝐽)) → (∀𝑦 ∈ (𝐽 qTop (𝐹 𝐽))((𝐹 𝐽) “ 𝑦) ∈ 𝐽 → ∀𝑦𝑥 ((𝐹 𝐽) “ 𝑦) ∈ 𝐽))
2826, 27mpan9 506 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ 𝑥 ⊆ (𝐽 qTop (𝐹 𝐽))) → ∀𝑦𝑥 ((𝐹 𝐽) “ 𝑦) ∈ 𝐽)
29 iunopn 22904 . . . . . . . . 9 ((𝐽 ∈ Top ∧ ∀𝑦𝑥 ((𝐹 𝐽) “ 𝑦) ∈ 𝐽) → 𝑦𝑥 ((𝐹 𝐽) “ 𝑦) ∈ 𝐽)
304, 28, 29syl2an2r 685 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ 𝑥 ⊆ (𝐽 qTop (𝐹 𝐽))) → 𝑦𝑥 ((𝐹 𝐽) “ 𝑦) ∈ 𝐽)
3124, 30eqeltrid 2845 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ 𝑥 ⊆ (𝐽 qTop (𝐹 𝐽))) → ((𝐹 𝐽) “ 𝑥) ∈ 𝐽)
3231ex 412 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝑥 ⊆ (𝐽 qTop (𝐹 𝐽)) → ((𝐹 𝐽) “ 𝑥) ∈ 𝐽))
3323, 32jcad 512 . . . . 5 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝑥 ⊆ (𝐽 qTop (𝐹 𝐽)) → ( 𝑥 ⊆ ran (𝐹 𝐽) ∧ ((𝐹 𝐽) “ 𝑥) ∈ 𝐽)))
341elqtop 23705 . . . . . 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 1373 . . . . 5 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → ( 𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ↔ ( 𝑥 ⊆ ran (𝐹 𝐽) ∧ ((𝐹 𝐽) “ 𝑥) ∈ 𝐽)))
3633, 35sylibrd 259 . . . 4 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝑥 ⊆ (𝐽 qTop (𝐹 𝐽)) → 𝑥 ∈ (𝐽 qTop (𝐹 𝐽))))
3736alrimiv 1927 . . 3 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → ∀𝑥(𝑥 ⊆ (𝐽 qTop (𝐹 𝐽)) → 𝑥 ∈ (𝐽 qTop (𝐹 𝐽))))
38 inss1 4237 . . . . . 6 (𝑥𝑦) ⊆ 𝑥
391elqtop 23705 . . . . . . . . . 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 1373 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ↔ (𝑥 ⊆ ran (𝐹 𝐽) ∧ ((𝐹 𝐽) “ 𝑥) ∈ 𝐽)))
4140biimpa 476 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ 𝑥 ∈ (𝐽 qTop (𝐹 𝐽))) → (𝑥 ⊆ ran (𝐹 𝐽) ∧ ((𝐹 𝐽) “ 𝑥) ∈ 𝐽))
4241adantrr 717 . . . . . . 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 494 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → 𝑥 ⊆ ran (𝐹 𝐽))
4438, 43sstrid 3995 . . . . 5 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → (𝑥𝑦) ⊆ ran (𝐹 𝐽))
456adantr 480 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → Fun (𝐹 𝐽))
46 inpreima 7084 . . . . . . 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 480 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → 𝐽 ∈ Top)
4942simprd 495 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → ((𝐹 𝐽) “ 𝑥) ∈ 𝐽)
5025adantrl 716 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → ((𝐹 𝐽) “ 𝑦) ∈ 𝐽)
51 inopn 22905 . . . . . . 7 ((𝐽 ∈ Top ∧ ((𝐹 𝐽) “ 𝑥) ∈ 𝐽 ∧ ((𝐹 𝐽) “ 𝑦) ∈ 𝐽) → (((𝐹 𝐽) “ 𝑥) ∩ ((𝐹 𝐽) “ 𝑦)) ∈ 𝐽)
5248, 49, 50, 51syl3anc 1373 . . . . . 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 2841 . . . . 5 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → ((𝐹 𝐽) “ (𝑥𝑦)) ∈ 𝐽)
541elqtop 23705 . . . . . . 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 1373 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → ((𝑥𝑦) ∈ (𝐽 qTop (𝐹 𝐽)) ↔ ((𝑥𝑦) ⊆ ran (𝐹 𝐽) ∧ ((𝐹 𝐽) “ (𝑥𝑦)) ∈ 𝐽)))
5655adantr 480 . . . . 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 713 . . . 4 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → (𝑥𝑦) ∈ (𝐽 qTop (𝐹 𝐽)))
5857ralrimivva 3202 . . 3 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → ∀𝑥 ∈ (𝐽 qTop (𝐹 𝐽))∀𝑦 ∈ (𝐽 qTop (𝐹 𝐽))(𝑥𝑦) ∈ (𝐽 qTop (𝐹 𝐽)))
59 ovex 7464 . . . 4 (𝐽 qTop (𝐹 𝐽)) ∈ V
60 istopg 22901 . . . 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 2841 1 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝐽 qTop 𝐹) ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087  wal 1538   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  cin 3950  wss 3951  𝒫 cpw 4600   cuni 4907   ciun 4991  ccnv 5684  dom cdm 5685  ran crn 5686  cres 5687  cima 5688  Fun wfun 6555  ontowfo 6559  (class class class)co 7431   qTop cqtop 17548  Topctop 22899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-qtop 17552  df-top 22900
This theorem is referenced by:  qtoptop  23708
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