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Theorem qtoptop2 21782
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 2765 . . . 4 𝐽 = 𝐽
21qtopres 21781 . . 3 (𝐹𝑉 → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹 𝐽)))
323ad2ant2 1164 . 2 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹 𝐽)))
4 simp1 1166 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → 𝐽 ∈ Top)
5 funres 6110 . . . . . . . . . . . . . . 15 (Fun 𝐹 → Fun (𝐹 𝐽))
653ad2ant3 1165 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → Fun (𝐹 𝐽))
7 funforn 6305 . . . . . . . . . . . . . 14 (Fun (𝐹 𝐽) ↔ (𝐹 𝐽):dom (𝐹 𝐽)–onto→ran (𝐹 𝐽))
86, 7sylib 209 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝐹 𝐽):dom (𝐹 𝐽)–onto→ran (𝐹 𝐽))
9 dmres 5594 . . . . . . . . . . . . . . 15 dom (𝐹 𝐽) = ( 𝐽 ∩ dom 𝐹)
10 inss1 3992 . . . . . . . . . . . . . . 15 ( 𝐽 ∩ dom 𝐹) ⊆ 𝐽
119, 10eqsstri 3795 . . . . . . . . . . . . . 14 dom (𝐹 𝐽) ⊆ 𝐽
1211a1i 11 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → dom (𝐹 𝐽) ⊆ 𝐽)
131elqtop 21780 . . . . . . . . . . . . 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 1490 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝑦 ∈ (𝐽 qTop (𝐹 𝐽)) ↔ (𝑦 ⊆ ran (𝐹 𝐽) ∧ ((𝐹 𝐽) “ 𝑦) ∈ 𝐽)))
1514simprbda 492 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽))) → 𝑦 ⊆ ran (𝐹 𝐽))
16 selpw 4322 . . . . . . . . . . 11 (𝑦 ∈ 𝒫 ran (𝐹 𝐽) ↔ 𝑦 ⊆ ran (𝐹 𝐽))
1715, 16sylibr 225 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽))) → 𝑦 ∈ 𝒫 ran (𝐹 𝐽))
1817ex 401 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝑦 ∈ (𝐽 qTop (𝐹 𝐽)) → 𝑦 ∈ 𝒫 ran (𝐹 𝐽)))
1918ssrdv 3767 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝐽 qTop (𝐹 𝐽)) ⊆ 𝒫 ran (𝐹 𝐽))
20 sstr2 3768 . . . . . . . 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 4768 . . . . . . 7 (𝑥 ⊆ 𝒫 ran (𝐹 𝐽) ↔ 𝑥 ⊆ ran (𝐹 𝐽))
2321, 22syl6ib 242 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝑥 ⊆ (𝐽 qTop (𝐹 𝐽)) → 𝑥 ⊆ ran (𝐹 𝐽)))
24 imauni 6696 . . . . . . . 8 ((𝐹 𝐽) “ 𝑥) = 𝑦𝑥 ((𝐹 𝐽) “ 𝑦)
25 simpl1 1242 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ 𝑥 ⊆ (𝐽 qTop (𝐹 𝐽))) → 𝐽 ∈ Top)
2614simplbda 493 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽))) → ((𝐹 𝐽) “ 𝑦) ∈ 𝐽)
2726ralrimiva 3113 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → ∀𝑦 ∈ (𝐽 qTop (𝐹 𝐽))((𝐹 𝐽) “ 𝑦) ∈ 𝐽)
28 ssralv 3826 . . . . . . . . . 10 (𝑥 ⊆ (𝐽 qTop (𝐹 𝐽)) → (∀𝑦 ∈ (𝐽 qTop (𝐹 𝐽))((𝐹 𝐽) “ 𝑦) ∈ 𝐽 → ∀𝑦𝑥 ((𝐹 𝐽) “ 𝑦) ∈ 𝐽))
2927, 28mpan9 502 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ 𝑥 ⊆ (𝐽 qTop (𝐹 𝐽))) → ∀𝑦𝑥 ((𝐹 𝐽) “ 𝑦) ∈ 𝐽)
30 iunopn 20982 . . . . . . . . 9 ((𝐽 ∈ Top ∧ ∀𝑦𝑥 ((𝐹 𝐽) “ 𝑦) ∈ 𝐽) → 𝑦𝑥 ((𝐹 𝐽) “ 𝑦) ∈ 𝐽)
3125, 29, 30syl2anc 579 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ 𝑥 ⊆ (𝐽 qTop (𝐹 𝐽))) → 𝑦𝑥 ((𝐹 𝐽) “ 𝑦) ∈ 𝐽)
3224, 31syl5eqel 2848 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ 𝑥 ⊆ (𝐽 qTop (𝐹 𝐽))) → ((𝐹 𝐽) “ 𝑥) ∈ 𝐽)
3332ex 401 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝑥 ⊆ (𝐽 qTop (𝐹 𝐽)) → ((𝐹 𝐽) “ 𝑥) ∈ 𝐽))
3423, 33jcad 508 . . . . 5 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝑥 ⊆ (𝐽 qTop (𝐹 𝐽)) → ( 𝑥 ⊆ ran (𝐹 𝐽) ∧ ((𝐹 𝐽) “ 𝑥) ∈ 𝐽)))
351elqtop 21780 . . . . . 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 1490 . . . . 5 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → ( 𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ↔ ( 𝑥 ⊆ ran (𝐹 𝐽) ∧ ((𝐹 𝐽) “ 𝑥) ∈ 𝐽)))
3734, 36sylibrd 250 . . . 4 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝑥 ⊆ (𝐽 qTop (𝐹 𝐽)) → 𝑥 ∈ (𝐽 qTop (𝐹 𝐽))))
3837alrimiv 2022 . . 3 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → ∀𝑥(𝑥 ⊆ (𝐽 qTop (𝐹 𝐽)) → 𝑥 ∈ (𝐽 qTop (𝐹 𝐽))))
39 inss1 3992 . . . . . 6 (𝑥𝑦) ⊆ 𝑥
401elqtop 21780 . . . . . . . . . 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 1490 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ↔ (𝑥 ⊆ ran (𝐹 𝐽) ∧ ((𝐹 𝐽) “ 𝑥) ∈ 𝐽)))
4241biimpa 468 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ 𝑥 ∈ (𝐽 qTop (𝐹 𝐽))) → (𝑥 ⊆ ran (𝐹 𝐽) ∧ ((𝐹 𝐽) “ 𝑥) ∈ 𝐽))
4342adantrr 708 . . . . . . 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 488 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → 𝑥 ⊆ ran (𝐹 𝐽))
4539, 44syl5ss 3772 . . . . 5 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → (𝑥𝑦) ⊆ ran (𝐹 𝐽))
466adantr 472 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → Fun (𝐹 𝐽))
47 inpreima 6532 . . . . . . 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 472 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → 𝐽 ∈ Top)
5043simprd 489 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → ((𝐹 𝐽) “ 𝑥) ∈ 𝐽)
5126adantrl 707 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → ((𝐹 𝐽) “ 𝑦) ∈ 𝐽)
52 inopn 20983 . . . . . . 7 ((𝐽 ∈ Top ∧ ((𝐹 𝐽) “ 𝑥) ∈ 𝐽 ∧ ((𝐹 𝐽) “ 𝑦) ∈ 𝐽) → (((𝐹 𝐽) “ 𝑥) ∩ ((𝐹 𝐽) “ 𝑦)) ∈ 𝐽)
5349, 50, 51, 52syl3anc 1490 . . . . . 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 2844 . . . . 5 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → ((𝐹 𝐽) “ (𝑥𝑦)) ∈ 𝐽)
551elqtop 21780 . . . . . . 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 1490 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → ((𝑥𝑦) ∈ (𝐽 qTop (𝐹 𝐽)) ↔ ((𝑥𝑦) ⊆ ran (𝐹 𝐽) ∧ ((𝐹 𝐽) “ (𝑥𝑦)) ∈ 𝐽)))
5756adantr 472 . . . . 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 704 . . . 4 (((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 𝐽)))) → (𝑥𝑦) ∈ (𝐽 qTop (𝐹 𝐽)))
5958ralrimivva 3118 . . 3 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → ∀𝑥 ∈ (𝐽 qTop (𝐹 𝐽))∀𝑦 ∈ (𝐽 qTop (𝐹 𝐽))(𝑥𝑦) ∈ (𝐽 qTop (𝐹 𝐽)))
60 ovex 6874 . . . 4 (𝐽 qTop (𝐹 𝐽)) ∈ V
61 istopg 20979 . . . 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 578 . 2 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝐽 qTop (𝐹 𝐽)) ∈ Top)
643, 63eqeltrd 2844 1 ((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝐽 qTop 𝐹) ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107  wal 1650   = wceq 1652  wcel 2155  wral 3055  Vcvv 3350  cin 3731  wss 3732  𝒫 cpw 4315   cuni 4594   ciun 4676  ccnv 5276  dom cdm 5277  ran crn 5278  cres 5279  cima 5280  Fun wfun 6062  ontowfo 6066  (class class class)co 6842   qTop cqtop 16431  Topctop 20977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-qtop 16435  df-top 20978
This theorem is referenced by:  qtoptop  21783
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