Theorem qtopcld 22318

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 22309	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
2		topontop 21518	. . 3 ⊢ ((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) → (𝐽 qTop 𝐹) ∈ Top)
3		eqid 2798	. . . 4 ⊢ ∪ (𝐽 qTop 𝐹) = ∪ (𝐽 qTop 𝐹)
4	3	iscld 21632	. . 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 21519	. . . . 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 3947	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐴 ⊆ 𝑌 ↔ 𝐴 ⊆ ∪ (𝐽 qTop 𝐹)))
9	7	difeq1d 4049	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝑌 ∖ 𝐴) = (∪ (𝐽 qTop 𝐹) ∖ 𝐴))
10	9	eleq1d 2874	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → ((𝑌 ∖ 𝐴) ∈ (𝐽 qTop 𝐹) ↔ (∪ (𝐽 qTop 𝐹) ∖ 𝐴) ∈ (𝐽 qTop 𝐹)))
11	8, 10	anbi12d 633	. 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 22308	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → ((𝑌 ∖ 𝐴) ∈ (𝐽 qTop 𝐹) ↔ ((𝑌 ∖ 𝐴) ⊆ 𝑌 ∧ (◡𝐹 “ (𝑌 ∖ 𝐴)) ∈ 𝐽)))
13	12	adantr 484	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → ((𝑌 ∖ 𝐴) ∈ (𝐽 qTop 𝐹) ↔ ((𝑌 ∖ 𝐴) ⊆ 𝑌 ∧ (◡𝐹 “ (𝑌 ∖ 𝐴)) ∈ 𝐽)))
14		difss 4059	. . . . . 6 ⊢ (𝑌 ∖ 𝐴) ⊆ 𝑌
15	14	biantrur 534	. . . . 5 ⊢ ((◡𝐹 “ (𝑌 ∖ 𝐴)) ∈ 𝐽 ↔ ((𝑌 ∖ 𝐴) ⊆ 𝑌 ∧ (◡𝐹 “ (𝑌 ∖ 𝐴)) ∈ 𝐽))
16		fofun 6566	. . . . . . . . . 10 ⊢ (𝐹:𝑋–onto→𝑌 → Fun 𝐹)
17	16	ad2antlr 726	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → Fun 𝐹)
18		funcnvcnv 6391	. . . . . . . . 9 ⊢ (Fun 𝐹 → Fun ◡◡𝐹)
19		imadif 6408	. . . . . . . . 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 6565	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌)
22		fimacnv 6816	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋⟶𝑌 → (◡𝐹 “ 𝑌) = 𝑋)
23	21, 22	syl 17	. . . . . . . . . . 11 ⊢ (𝐹:𝑋–onto→𝑌 → (◡𝐹 “ 𝑌) = 𝑋)
24	23	ad2antlr 726	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → (◡𝐹 “ 𝑌) = 𝑋)
25		toponuni 21519	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
26	25	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → 𝑋 = ∪ 𝐽)
27	24, 26	eqtrd 2833	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → (◡𝐹 “ 𝑌) = ∪ 𝐽)
28	27	difeq1d 4049	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → ((◡𝐹 “ 𝑌) ∖ (◡𝐹 “ 𝐴)) = (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)))
29	20, 28	eqtrd 2833	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → (◡𝐹 “ (𝑌 ∖ 𝐴)) = (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)))
30	29	eleq1d 2874	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → ((◡𝐹 “ (𝑌 ∖ 𝐴)) ∈ 𝐽 ↔ (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)) ∈ 𝐽))
31		topontop 21518	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
32	31	ad2antrr 725	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → 𝐽 ∈ Top)
33		cnvimass 5916	. . . . . . . . 9 ⊢ (◡𝐹 “ 𝐴) ⊆ dom 𝐹
34		fofn 6567	. . . . . . . . . . 11 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹 Fn 𝑋)
35	34	fndmd 6427	. . . . . . . . . 10 ⊢ (𝐹:𝑋–onto→𝑌 → dom 𝐹 = 𝑋)
36	35	ad2antlr 726	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → dom 𝐹 = 𝑋)
37	33, 36	sseqtrid 3967	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → (◡𝐹 “ 𝐴) ⊆ 𝑋)
38	37, 26	sseqtrd 3955	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → (◡𝐹 “ 𝐴) ⊆ ∪ 𝐽)
39		eqid 2798	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
40	39	iscld2 21633	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (◡𝐹 “ 𝐴) ⊆ ∪ 𝐽) → ((◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)) ∈ 𝐽))
41	32, 38, 40	syl2anc 587	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → ((◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)) ∈ 𝐽))
42	30, 41	bitr4d 285	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → ((◡𝐹 “ (𝑌 ∖ 𝐴)) ∈ 𝐽 ↔ (◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽)))
43	15, 42	bitr3id 288	. . . 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 282	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → ((𝑌 ∖ 𝐴) ∈ (𝐽 qTop 𝐹) ↔ (◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽)))
45	44	pm5.32da 582	. 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 310	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐴 ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ (𝐴 ⊆ 𝑌 ∧ (◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ∖ cdif 3878   ⊆ wss 3881  ∪ cuni 4800  ^◡ccnv 5518  dom cdm 5519   “ cima 5522  Fun wfun 6318  ⟶wf 6320  –onto→wfo 6322  ‘cfv 6324  (class class class)co 7135   qTop cqtop 16768  Topctop 21498  TopOnctopon 21515  Clsdccld 21621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-qtop 16772  df-top 21499  df-topon 21516  df-cld 21624

This theorem is referenced by:  qtoprest  22322  kqcld  22340  qustgphaus  22728  qtopt1  31188