Theorem qtoptopon 21836

Description: The base set of the quotient topology. (Contributed by Mario Carneiro, 22-Aug-2015.)

Assertion

Ref	Expression
qtoptopon	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))

Proof of Theorem qtoptopon

Step	Hyp	Ref	Expression
1		toponuni 21047	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
2		foeq2 6328	. . . . . 6 ⊢ (𝑋 = ∪ 𝐽 → (𝐹:𝑋–onto→𝑌 ↔ 𝐹:∪ 𝐽–onto→𝑌))
3	1, 2	syl 17	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐹:𝑋–onto→𝑌 ↔ 𝐹:∪ 𝐽–onto→𝑌))
4	3	biimpa 469	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → 𝐹:∪ 𝐽–onto→𝑌)
5		fofn 6333	. . . 4 ⊢ (𝐹:∪ 𝐽–onto→𝑌 → 𝐹 Fn ∪ 𝐽)
6	4, 5	syl 17	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → 𝐹 Fn ∪ 𝐽)
7		topontop 21046	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
8		eqid 2799	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
9	8	qtoptop 21832	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹 Fn ∪ 𝐽) → (𝐽 qTop 𝐹) ∈ Top)
10	7, 9	sylan 576	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn ∪ 𝐽) → (𝐽 qTop 𝐹) ∈ Top)
11	6, 10	syldan 586	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐽 qTop 𝐹) ∈ Top)
12	8	qtopuni 21834	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹:∪ 𝐽–onto→𝑌) → 𝑌 = ∪ (𝐽 qTop 𝐹))
13	7, 12	sylan 576	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:∪ 𝐽–onto→𝑌) → 𝑌 = ∪ (𝐽 qTop 𝐹))
14	4, 13	syldan 586	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → 𝑌 = ∪ (𝐽 qTop 𝐹))
15		istopon 21045	. 2 ⊢ ((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) ↔ ((𝐽 qTop 𝐹) ∈ Top ∧ 𝑌 = ∪ (𝐽 qTop 𝐹)))
16	11, 14, 15	sylanbrc 579	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385   = wceq 1653   ∈ wcel 2157  ∪ cuni 4628   Fn wfn 6096  –onto→wfo 6099  ‘cfv 6101  (class class class)co 6878   qTop cqtop 16478  Topctop 21026  TopOnctopon 21043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-qtop 16482  df-top 21027  df-topon 21044

This theorem is referenced by:  qtopid  21837  qtopcld  21845  qtopcn  21846  qtopeu  21848  qtoprest  21849  imastps  21853  kqtopon  21859  qtopf1  21948  qtophmeo  21949  qustgplem  22252  qtophaus  30419