Theorem qtoptopon 21507

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 20719	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
2		foeq2 6112	. . . . . 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 501	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → 𝐹:∪ 𝐽–onto→𝑌)
5		fofn 6117	. . . 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 20718	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
8		eqid 2622	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
9	8	qtoptop 21503	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹 Fn ∪ 𝐽) → (𝐽 qTop 𝐹) ∈ Top)
10	7, 9	sylan 488	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn ∪ 𝐽) → (𝐽 qTop 𝐹) ∈ Top)
11	6, 10	syldan 487	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐽 qTop 𝐹) ∈ Top)
12	8	qtopuni 21505	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹:∪ 𝐽–onto→𝑌) → 𝑌 = ∪ (𝐽 qTop 𝐹))
13	7, 12	sylan 488	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:∪ 𝐽–onto→𝑌) → 𝑌 = ∪ (𝐽 qTop 𝐹))
14	4, 13	syldan 487	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → 𝑌 = ∪ (𝐽 qTop 𝐹))
15		istopon 20717	. 2 ⊢ ((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) ↔ ((𝐽 qTop 𝐹) ∈ Top ∧ 𝑌 = ∪ (𝐽 qTop 𝐹)))
16	11, 14, 15	sylanbrc 698	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∪ cuni 4436   Fn wfn 5883  –onto→wfo 5886  ‘cfv 5888  (class class class)co 6650   qTop cqtop 16163  Topctop 20698  TopOnctopon 20715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-qtop 16167  df-top 20699  df-topon 20716

This theorem is referenced by:  qtopid  21508  qtopcld  21516  qtopcn  21517  qtopeu  21519  qtoprest  21520  imastps  21524  kqtopon  21530  qtopf1  21619  qtophmeo  21620  qustgplem  21924  qtophaus  29903