Theorem qtopcld 21516

Description: The property of being a closed set in the quotient topology. (Contributed by Mario Carneiro, 24-Mar-2015.)

Assertion

Ref	Expression
qtopcld	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐴 ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ (𝐴 ⊆ 𝑌 ∧ (◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽))))

Proof of Theorem qtopcld

Step	Hyp	Ref	Expression
1		qtoptopon 21507	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
2		topontop 20718	. . 3 ⊢ ((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) → (𝐽 qTop 𝐹) ∈ Top)
3		eqid 2622	. . . 4 ⊢ ∪ (𝐽 qTop 𝐹) = ∪ (𝐽 qTop 𝐹)
4	3	iscld 20831	. . 3 ⊢ ((𝐽 qTop 𝐹) ∈ Top → (𝐴 ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ (𝐴 ⊆ ∪ (𝐽 qTop 𝐹) ∧ (∪ (𝐽 qTop 𝐹) ∖ 𝐴) ∈ (𝐽 qTop 𝐹))))
5	1, 2, 4	3syl 18	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐴 ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ (𝐴 ⊆ ∪ (𝐽 qTop 𝐹) ∧ (∪ (𝐽 qTop 𝐹) ∖ 𝐴) ∈ (𝐽 qTop 𝐹))))
6		toponuni 20719	. . . . 5 ⊢ ((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) → 𝑌 = ∪ (𝐽 qTop 𝐹))
7	1, 6	syl 17	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → 𝑌 = ∪ (𝐽 qTop 𝐹))
8	7	sseq2d 3633	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐴 ⊆ 𝑌 ↔ 𝐴 ⊆ ∪ (𝐽 qTop 𝐹)))
9	7	difeq1d 3727	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝑌 ∖ 𝐴) = (∪ (𝐽 qTop 𝐹) ∖ 𝐴))
10	9	eleq1d 2686	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → ((𝑌 ∖ 𝐴) ∈ (𝐽 qTop 𝐹) ↔ (∪ (𝐽 qTop 𝐹) ∖ 𝐴) ∈ (𝐽 qTop 𝐹)))
11	8, 10	anbi12d 747	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → ((𝐴 ⊆ 𝑌 ∧ (𝑌 ∖ 𝐴) ∈ (𝐽 qTop 𝐹)) ↔ (𝐴 ⊆ ∪ (𝐽 qTop 𝐹) ∧ (∪ (𝐽 qTop 𝐹) ∖ 𝐴) ∈ (𝐽 qTop 𝐹))))
12		elqtop3 21506	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → ((𝑌 ∖ 𝐴) ∈ (𝐽 qTop 𝐹) ↔ ((𝑌 ∖ 𝐴) ⊆ 𝑌 ∧ (◡𝐹 “ (𝑌 ∖ 𝐴)) ∈ 𝐽)))
13	12	adantr 481	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → ((𝑌 ∖ 𝐴) ∈ (𝐽 qTop 𝐹) ↔ ((𝑌 ∖ 𝐴) ⊆ 𝑌 ∧ (◡𝐹 “ (𝑌 ∖ 𝐴)) ∈ 𝐽)))
14		difss 3737	. . . . . 6 ⊢ (𝑌 ∖ 𝐴) ⊆ 𝑌
15	14	biantrur 527	. . . . 5 ⊢ ((◡𝐹 “ (𝑌 ∖ 𝐴)) ∈ 𝐽 ↔ ((𝑌 ∖ 𝐴) ⊆ 𝑌 ∧ (◡𝐹 “ (𝑌 ∖ 𝐴)) ∈ 𝐽))
16		fofun 6116	. . . . . . . . . 10 ⊢ (𝐹:𝑋–onto→𝑌 → Fun 𝐹)
17	16	ad2antlr 763	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → Fun 𝐹)
18		funcnvcnv 5956	. . . . . . . . 9 ⊢ (Fun 𝐹 → Fun ◡◡𝐹)
19		imadif 5973	. . . . . . . . 9 ⊢ (Fun ◡◡𝐹 → (◡𝐹 “ (𝑌 ∖ 𝐴)) = ((◡𝐹 “ 𝑌) ∖ (◡𝐹 “ 𝐴)))
20	17, 18, 19	3syl 18	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → (◡𝐹 “ (𝑌 ∖ 𝐴)) = ((◡𝐹 “ 𝑌) ∖ (◡𝐹 “ 𝐴)))
21		fof 6115	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌)
22		fimacnv 6347	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋⟶𝑌 → (◡𝐹 “ 𝑌) = 𝑋)
23	21, 22	syl 17	. . . . . . . . . . 11 ⊢ (𝐹:𝑋–onto→𝑌 → (◡𝐹 “ 𝑌) = 𝑋)
24	23	ad2antlr 763	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → (◡𝐹 “ 𝑌) = 𝑋)
25		toponuni 20719	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
26	25	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → 𝑋 = ∪ 𝐽)
27	24, 26	eqtrd 2656	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → (◡𝐹 “ 𝑌) = ∪ 𝐽)
28	27	difeq1d 3727	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → ((◡𝐹 “ 𝑌) ∖ (◡𝐹 “ 𝐴)) = (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)))
29	20, 28	eqtrd 2656	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → (◡𝐹 “ (𝑌 ∖ 𝐴)) = (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)))
30	29	eleq1d 2686	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → ((◡𝐹 “ (𝑌 ∖ 𝐴)) ∈ 𝐽 ↔ (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)) ∈ 𝐽))
31		topontop 20718	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
32	31	ad2antrr 762	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → 𝐽 ∈ Top)
33		cnvimass 5485	. . . . . . . . 9 ⊢ (◡𝐹 “ 𝐴) ⊆ dom 𝐹
34		fofn 6117	. . . . . . . . . . 11 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹 Fn 𝑋)
35		fndm 5990	. . . . . . . . . . 11 ⊢ (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
36	34, 35	syl 17	. . . . . . . . . 10 ⊢ (𝐹:𝑋–onto→𝑌 → dom 𝐹 = 𝑋)
37	36	ad2antlr 763	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → dom 𝐹 = 𝑋)
38	33, 37	syl5sseq 3653	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → (◡𝐹 “ 𝐴) ⊆ 𝑋)
39	38, 26	sseqtrd 3641	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → (◡𝐹 “ 𝐴) ⊆ ∪ 𝐽)
40		eqid 2622	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
41	40	iscld2 20832	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (◡𝐹 “ 𝐴) ⊆ ∪ 𝐽) → ((◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)) ∈ 𝐽))
42	32, 39, 41	syl2anc 693	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → ((◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)) ∈ 𝐽))
43	30, 42	bitr4d 271	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → ((◡𝐹 “ (𝑌 ∖ 𝐴)) ∈ 𝐽 ↔ (◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽)))
44	15, 43	syl5bbr 274	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → (((𝑌 ∖ 𝐴) ⊆ 𝑌 ∧ (◡𝐹 “ (𝑌 ∖ 𝐴)) ∈ 𝐽) ↔ (◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽)))
45	13, 44	bitrd 268	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → ((𝑌 ∖ 𝐴) ∈ (𝐽 qTop 𝐹) ↔ (◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽)))
46	45	pm5.32da 673	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → ((𝐴 ⊆ 𝑌 ∧ (𝑌 ∖ 𝐴) ∈ (𝐽 qTop 𝐹)) ↔ (𝐴 ⊆ 𝑌 ∧ (◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽))))
47	5, 11, 46	3bitr2d 296	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐴 ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ (𝐴 ⊆ 𝑌 ∧ (◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽))))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ∖ cdif 3571   ⊆ wss 3574  ∪ cuni 4436  ^◡ccnv 5113  dom cdm 5114   “ cima 5117  Fun wfun 5882   Fn wfn 5883  ⟶wf 5884  –onto→wfo 5886  ‘cfv 5888  (class class class)co 6650   qTop cqtop 16163  Topctop 20698  TopOnctopon 20715  Clsdccld 20820

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  df-cld 20823

This theorem is referenced by:  qtoprest  21520  kqcld  21538  qustgphaus  21926  qtopt1  29902