Theorem qtopcld 22864

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 22855	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
2		topontop 22062	. . 3 ⊢ ((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) → (𝐽 qTop 𝐹) ∈ Top)
3		eqid 2738	. . . 4 ⊢ ∪ (𝐽 qTop 𝐹) = ∪ (𝐽 qTop 𝐹)
4	3	iscld 22178	. . 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 22063	. . . . 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 3953	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐴 ⊆ 𝑌 ↔ 𝐴 ⊆ ∪ (𝐽 qTop 𝐹)))
9	7	difeq1d 4056	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝑌 ∖ 𝐴) = (∪ (𝐽 qTop 𝐹) ∖ 𝐴))
10	9	eleq1d 2823	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → ((𝑌 ∖ 𝐴) ∈ (𝐽 qTop 𝐹) ↔ (∪ (𝐽 qTop 𝐹) ∖ 𝐴) ∈ (𝐽 qTop 𝐹)))
11	8, 10	anbi12d 631	. 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 22854	. . . . 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 4066	. . . . . 6 ⊢ (𝑌 ∖ 𝐴) ⊆ 𝑌
15	14	biantrur 531	. . . . 5 ⊢ ((◡𝐹 “ (𝑌 ∖ 𝐴)) ∈ 𝐽 ↔ ((𝑌 ∖ 𝐴) ⊆ 𝑌 ∧ (◡𝐹 “ (𝑌 ∖ 𝐴)) ∈ 𝐽))
16		fofun 6689	. . . . . . . . . 10 ⊢ (𝐹:𝑋–onto→𝑌 → Fun 𝐹)
17	16	ad2antlr 724	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → Fun 𝐹)
18		funcnvcnv 6501	. . . . . . . . 9 ⊢ (Fun 𝐹 → Fun ◡◡𝐹)
19		imadif 6518	. . . . . . . . 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 6688	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌)
22		fimacnv 6622	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋⟶𝑌 → (◡𝐹 “ 𝑌) = 𝑋)
23	21, 22	syl 17	. . . . . . . . . . 11 ⊢ (𝐹:𝑋–onto→𝑌 → (◡𝐹 “ 𝑌) = 𝑋)
24	23	ad2antlr 724	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → (◡𝐹 “ 𝑌) = 𝑋)
25		toponuni 22063	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
26	25	ad2antrr 723	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → 𝑋 = ∪ 𝐽)
27	24, 26	eqtrd 2778	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → (◡𝐹 “ 𝑌) = ∪ 𝐽)
28	27	difeq1d 4056	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → ((◡𝐹 “ 𝑌) ∖ (◡𝐹 “ 𝐴)) = (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)))
29	20, 28	eqtrd 2778	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → (◡𝐹 “ (𝑌 ∖ 𝐴)) = (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)))
30	29	eleq1d 2823	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → ((◡𝐹 “ (𝑌 ∖ 𝐴)) ∈ 𝐽 ↔ (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)) ∈ 𝐽))
31		topontop 22062	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
32	31	ad2antrr 723	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → 𝐽 ∈ Top)
33		cnvimass 5989	. . . . . . . . 9 ⊢ (◡𝐹 “ 𝐴) ⊆ dom 𝐹
34		fofn 6690	. . . . . . . . . . 11 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹 Fn 𝑋)
35	34	fndmd 6538	. . . . . . . . . 10 ⊢ (𝐹:𝑋–onto→𝑌 → dom 𝐹 = 𝑋)
36	35	ad2antlr 724	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → dom 𝐹 = 𝑋)
37	33, 36	sseqtrid 3973	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → (◡𝐹 “ 𝐴) ⊆ 𝑋)
38	37, 26	sseqtrd 3961	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → (◡𝐹 “ 𝐴) ⊆ ∪ 𝐽)
39		eqid 2738	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
40	39	iscld2 22179	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (◡𝐹 “ 𝐴) ⊆ ∪ 𝐽) → ((◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)) ∈ 𝐽))
41	32, 38, 40	syl2anc 584	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → ((◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)) ∈ 𝐽))
42	30, 41	bitr4d 281	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → ((◡𝐹 “ (𝑌 ∖ 𝐴)) ∈ 𝐽 ↔ (◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽)))
43	15, 42	bitr3id 285	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → (((𝑌 ∖ 𝐴) ⊆ 𝑌 ∧ (◡𝐹 “ (𝑌 ∖ 𝐴)) ∈ 𝐽) ↔ (◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽)))
44	13, 43	bitrd 278	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → ((𝑌 ∖ 𝐴) ∈ (𝐽 qTop 𝐹) ↔ (◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽)))
45	44	pm5.32da 579	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → ((𝐴 ⊆ 𝑌 ∧ (𝑌 ∖ 𝐴) ∈ (𝐽 qTop 𝐹)) ↔ (𝐴 ⊆ 𝑌 ∧ (◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽))))
46	5, 11, 45	3bitr2d 307	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐴 ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ (𝐴 ⊆ 𝑌 ∧ (◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ∖ cdif 3884   ⊆ wss 3887  ∪ cuni 4839  ^◡ccnv 5588  dom cdm 5589   “ cima 5592  Fun wfun 6427  ⟶wf 6429  –onto→wfo 6431  ‘cfv 6433  (class class class)co 7275   qTop cqtop 17214  Topctop 22042  TopOnctopon 22059  Clsdccld 22167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-qtop 17218  df-top 22043  df-topon 22060  df-cld 22170

This theorem is referenced by:  qtoprest  22868  kqcld  22886  qustgphaus  23274  qtopt1  31785