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Theorem qtoptop2 21722
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 2770 . . . 4 𝐽 = 𝐽
21qtopres 21721 . . 3 (𝐹𝑉 → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹 𝐽)))
323ad2ant2 1127 . 2 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹 𝐽)))
4 simp1 1129 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → 𝐽 ∈ Top)
5 funres 6072 . . . . . . . . . . . . . . 15 (Fun 𝐹 → Fun (𝐹 𝐽))
653ad2ant3 1128 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → Fun (𝐹 𝐽))
7 funforn 6263 . . . . . . . . . . . . . 14 (Fun (𝐹 𝐽) ↔ (𝐹 𝐽):dom (𝐹 𝐽)–onto→ran (𝐹 𝐽))
86, 7sylib 208 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝐹 𝐽):dom (𝐹 𝐽)–onto→ran (𝐹 𝐽))
9 dmres 5560 . . . . . . . . . . . . . . 15 dom (𝐹 𝐽) = ( 𝐽 ∩ dom 𝐹)
10 inss1 3979 . . . . . . . . . . . . . . 15 ( 𝐽 ∩ dom 𝐹) ⊆ 𝐽
119, 10eqsstri 3782 . . . . . . . . . . . . . 14 dom (𝐹 𝐽) ⊆ 𝐽
1211a1i 11 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → dom (𝐹 𝐽) ⊆ 𝐽)
131elqtop 21720 . . . . . . . . . . . . 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 1475 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝑦 ∈ (𝐽 qTop (𝐹 𝐽)) ↔ (𝑦 ⊆ ran (𝐹 𝐽) ∧ ((𝐹 𝐽) “ 𝑦) ∈ 𝐽)))
1514simprbda 480 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽))) → 𝑦 ⊆ ran (𝐹 𝐽))
16 selpw 4302 . . . . . . . . . . 11 (𝑦 ∈ 𝒫 ran (𝐹 𝐽) ↔ 𝑦 ⊆ ran (𝐹 𝐽))
1715, 16sylibr 224 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽))) → 𝑦 ∈ 𝒫 ran (𝐹 𝐽))
1817ex 397 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝑦 ∈ (𝐽 qTop (𝐹 𝐽)) → 𝑦 ∈ 𝒫 ran (𝐹 𝐽)))
1918ssrdv 3756 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝐽 qTop (𝐹 𝐽)) ⊆ 𝒫 ran (𝐹 𝐽))
20 sstr2 3757 . . . . . . . 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 4743 . . . . . . 7 (𝑥 ⊆ 𝒫 ran (𝐹 𝐽) ↔ 𝑥 ⊆ ran (𝐹 𝐽))
2321, 22syl6ib 241 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝑥 ⊆ (𝐽 qTop (𝐹 𝐽)) → 𝑥 ⊆ ran (𝐹 𝐽)))
24 imauni 6646 . . . . . . . 8 ((𝐹 𝐽) “ 𝑥) = 𝑦𝑥 ((𝐹 𝐽) “ 𝑦)
25 simpl1 1226 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ 𝑥 ⊆ (𝐽 qTop (𝐹 𝐽))) → 𝐽 ∈ Top)
2614simplbda 481 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽))) → ((𝐹 𝐽) “ 𝑦) ∈ 𝐽)
2726ralrimiva 3114 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → ∀𝑦 ∈ (𝐽 qTop (𝐹 𝐽))((𝐹 𝐽) “ 𝑦) ∈ 𝐽)
28 ssralv 3813 . . . . . . . . . 10 (𝑥 ⊆ (𝐽 qTop (𝐹 𝐽)) → (∀𝑦 ∈ (𝐽 qTop (𝐹 𝐽))((𝐹 𝐽) “ 𝑦) ∈ 𝐽 → ∀𝑦𝑥 ((𝐹 𝐽) “ 𝑦) ∈ 𝐽))
2927, 28mpan9 490 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ 𝑥 ⊆ (𝐽 qTop (𝐹 𝐽))) → ∀𝑦𝑥 ((𝐹 𝐽) “ 𝑦) ∈ 𝐽)
30 iunopn 20922 . . . . . . . . 9 ((𝐽 ∈ Top ∧ ∀𝑦𝑥 ((𝐹 𝐽) “ 𝑦) ∈ 𝐽) → 𝑦𝑥 ((𝐹 𝐽) “ 𝑦) ∈ 𝐽)
3125, 29, 30syl2anc 565 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ 𝑥 ⊆ (𝐽 qTop (𝐹 𝐽))) → 𝑦𝑥 ((𝐹 𝐽) “ 𝑦) ∈ 𝐽)
3224, 31syl5eqel 2853 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ 𝑥 ⊆ (𝐽 qTop (𝐹 𝐽))) → ((𝐹 𝐽) “ 𝑥) ∈ 𝐽)
3332ex 397 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝑥 ⊆ (𝐽 qTop (𝐹 𝐽)) → ((𝐹 𝐽) “ 𝑥) ∈ 𝐽))
3423, 33jcad 496 . . . . 5 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝑥 ⊆ (𝐽 qTop (𝐹 𝐽)) → ( 𝑥 ⊆ ran (𝐹 𝐽) ∧ ((𝐹 𝐽) “ 𝑥) ∈ 𝐽)))
351elqtop 21720 . . . . . 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)))
364, 8, 12, 35syl3anc 1475 . . . . 5 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → ( 𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ↔ ( 𝑥 ⊆ ran (𝐹 𝐽) ∧ ((𝐹 𝐽) “ 𝑥) ∈ 𝐽)))
3734, 36sylibrd 249 . . . 4 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝑥 ⊆ (𝐽 qTop (𝐹 𝐽)) → 𝑥 ∈ (𝐽 qTop (𝐹 𝐽))))
3837alrimiv 2006 . . 3 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → ∀𝑥(𝑥 ⊆ (𝐽 qTop (𝐹 𝐽)) → 𝑥 ∈ (𝐽 qTop (𝐹 𝐽))))
39 inss1 3979 . . . . . 6 (𝑥𝑦) ⊆ 𝑥
401elqtop 21720 . . . . . . . . . 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)))
414, 8, 12, 40syl3anc 1475 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ↔ (𝑥 ⊆ ran (𝐹 𝐽) ∧ ((𝐹 𝐽) “ 𝑥) ∈ 𝐽)))
4241biimpa 462 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ 𝑥 ∈ (𝐽 qTop (𝐹 𝐽))) → (𝑥 ⊆ ran (𝐹 𝐽) ∧ ((𝐹 𝐽) “ 𝑥) ∈ 𝐽))
4342adantrr 688 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → (𝑥 ⊆ ran (𝐹 𝐽) ∧ ((𝐹 𝐽) “ 𝑥) ∈ 𝐽))
4443simpld 476 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → 𝑥 ⊆ ran (𝐹 𝐽))
4539, 44syl5ss 3761 . . . . 5 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → (𝑥𝑦) ⊆ ran (𝐹 𝐽))
466adantr 466 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → Fun (𝐹 𝐽))
47 inpreima 6485 . . . . . . 7 (Fun (𝐹 𝐽) → ((𝐹 𝐽) “ (𝑥𝑦)) = (((𝐹 𝐽) “ 𝑥) ∩ ((𝐹 𝐽) “ 𝑦)))
4846, 47syl 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)))
494adantr 466 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → 𝐽 ∈ Top)
5043simprd 477 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → ((𝐹 𝐽) “ 𝑥) ∈ 𝐽)
5126adantrl 687 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → ((𝐹 𝐽) “ 𝑦) ∈ 𝐽)
52 inopn 20923 . . . . . . 7 ((𝐽 ∈ Top ∧ ((𝐹 𝐽) “ 𝑥) ∈ 𝐽 ∧ ((𝐹 𝐽) “ 𝑦) ∈ 𝐽) → (((𝐹 𝐽) “ 𝑥) ∩ ((𝐹 𝐽) “ 𝑦)) ∈ 𝐽)
5349, 50, 51, 52syl3anc 1475 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → (((𝐹 𝐽) “ 𝑥) ∩ ((𝐹 𝐽) “ 𝑦)) ∈ 𝐽)
5448, 53eqeltrd 2849 . . . . 5 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → ((𝐹 𝐽) “ (𝑥𝑦)) ∈ 𝐽)
551elqtop 21720 . . . . . . 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)))
564, 8, 12, 55syl3anc 1475 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → ((𝑥𝑦) ∈ (𝐽 qTop (𝐹 𝐽)) ↔ ((𝑥𝑦) ⊆ ran (𝐹 𝐽) ∧ ((𝐹 𝐽) “ (𝑥𝑦)) ∈ 𝐽)))
5756adantr 466 . . . . 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)))
5845, 54, 57mpbir2and 684 . . . 4 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → (𝑥𝑦) ∈ (𝐽 qTop (𝐹 𝐽)))
5958ralrimivva 3119 . . 3 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → ∀𝑥 ∈ (𝐽 qTop (𝐹 𝐽))∀𝑦 ∈ (𝐽 qTop (𝐹 𝐽))(𝑥𝑦) ∈ (𝐽 qTop (𝐹 𝐽)))
60 ovex 6822 . . . 4 (𝐽 qTop (𝐹 𝐽)) ∈ V
61 istopg 20919 . . . 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)))))
6260, 61ax-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))))
6338, 59, 62sylanbrc 564 . 2 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝐽 qTop (𝐹 𝐽)) ∈ Top)
643, 63eqeltrd 2849 1 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝐽 qTop 𝐹) ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1070  wal 1628   = wceq 1630  wcel 2144  wral 3060  Vcvv 3349  cin 3720  wss 3721  𝒫 cpw 4295   cuni 4572   ciun 4652  ccnv 5248  dom cdm 5249  ran crn 5250  cres 5251  cima 5252  Fun wfun 6025  ontowfo 6029  (class class class)co 6792   qTop cqtop 16370  Topctop 20917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-qtop 16374  df-top 20918
This theorem is referenced by:  qtoptop  21723
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