Theorem qtopcld 24087

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⟶onto𝑌) → (𝐴 ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ (𝐴 ⊆ 𝑌 ∧ (◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽))))

Proof of Theorem qtopcld

Step	Hyp	Ref	Expression
1		qtoptopon 24078	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶onto𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
2		topontop 23286	. . 3 ⊢ ((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) → (𝐽 qTop 𝐹) ∈ Top)
3		eqid 2763	. . . 4 ⊢ ∪ (𝐽 qTop 𝐹) = ∪ (𝐽 qTop 𝐹)
4	3	iscld 23400	. . 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 19	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶onto𝑌) → (𝐴 ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ (𝐴 ⊆ ∪ (𝐽 qTop 𝐹) ∧ (∪ (𝐽 qTop 𝐹) ∖ 𝐴) ∈ (𝐽 qTop 𝐹))))
6		toponuni 23287	. . . . 5 ⊢ ((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) → 𝑌 = ∪ (𝐽 qTop 𝐹))
7	1, 6	syl 18	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶onto𝑌) → 𝑌 = ∪ (𝐽 qTop 𝐹))
8	7	sseq2d 3945	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶onto𝑌) → (𝐴 ⊆ 𝑌 ↔ 𝐴 ⊆ ∪ (𝐽 qTop 𝐹)))
9	7	difeq1d 4065	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶onto𝑌) → (𝑌 ∖ 𝐴) = (∪ (𝐽 qTop 𝐹) ∖ 𝐴))
10	9	eleq1d 2849	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶onto𝑌) → ((𝑌 ∖ 𝐴) ∈ (𝐽 qTop 𝐹) ↔ (∪ (𝐽 qTop 𝐹) ∖ 𝐴) ∈ (𝐽 qTop 𝐹)))
11	8, 10	anbi12d 644	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶onto𝑌) → ((𝐴 ⊆ 𝑌 ∧ (𝑌 ∖ 𝐴) ∈ (𝐽 qTop 𝐹)) ↔ (𝐴 ⊆ ∪ (𝐽 qTop 𝐹) ∧ (∪ (𝐽 qTop 𝐹) ∖ 𝐴) ∈ (𝐽 qTop 𝐹))))
12		elqtop3 24077	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶onto𝑌) → ((𝑌 ∖ 𝐴) ∈ (𝐽 qTop 𝐹) ↔ ((𝑌 ∖ 𝐴) ⊆ 𝑌 ∧ (◡𝐹 “ (𝑌 ∖ 𝐴)) ∈ 𝐽)))
13	12	adantr 486	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶onto𝑌) ∧ 𝐴 ⊆ 𝑌) → ((𝑌 ∖ 𝐴) ∈ (𝐽 qTop 𝐹) ↔ ((𝑌 ∖ 𝐴) ⊆ 𝑌 ∧ (◡𝐹 “ (𝑌 ∖ 𝐴)) ∈ 𝐽)))
14		difss 4075	. . . . . 6 ⊢ (𝑌 ∖ 𝐴) ⊆ 𝑌
15	14	biantrur 540	. . . . 5 ⊢ ((◡𝐹 “ (𝑌 ∖ 𝐴)) ∈ 𝐽 ↔ ((𝑌 ∖ 𝐴) ⊆ 𝑌 ∧ (◡𝐹 “ (𝑌 ∖ 𝐴)) ∈ 𝐽))
16		fofun 6827	. . . . . . . . . 10 ⊢ (𝐹:𝑋⟶onto𝑌 → Fun 𝐹)
17	16	ad2antlr 740	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶onto𝑌) ∧ 𝐴 ⊆ 𝑌) → Fun 𝐹)
18		funcnvcnv 6637	. . . . . . . . 9 ⊢ (Fun 𝐹 → Fun ◡◡𝐹)
19		imadif 6654	. . . . . . . . 9 ⊢ (Fun ◡◡𝐹 → (◡𝐹 “ (𝑌 ∖ 𝐴)) = ((◡𝐹 “ 𝑌) ∖ (◡𝐹 “ 𝐴)))
20	17, 18, 19	3syl 19	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶onto𝑌) ∧ 𝐴 ⊆ 𝑌) → (◡𝐹 “ (𝑌 ∖ 𝐴)) = ((◡𝐹 “ 𝑌) ∖ (◡𝐹 “ 𝐴)))
21		fof 6826	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋⟶onto𝑌 → 𝐹:𝑋⟶𝑌)
22		fimacnv 6762	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋⟶𝑌 → (◡𝐹 “ 𝑌) = 𝑋)
23	21, 22	syl 18	. . . . . . . . . . 11 ⊢ (𝐹:𝑋⟶onto𝑌 → (◡𝐹 “ 𝑌) = 𝑋)
24	23	ad2antlr 740	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶onto𝑌) ∧ 𝐴 ⊆ 𝑌) → (◡𝐹 “ 𝑌) = 𝑋)
25		toponuni 23287	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
26	25	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶onto𝑌) ∧ 𝐴 ⊆ 𝑌) → 𝑋 = ∪ 𝐽)
27	24, 26	eqtrd 2798	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶onto𝑌) ∧ 𝐴 ⊆ 𝑌) → (◡𝐹 “ 𝑌) = ∪ 𝐽)
28	27	difeq1d 4065	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶onto𝑌) ∧ 𝐴 ⊆ 𝑌) → ((◡𝐹 “ 𝑌) ∖ (◡𝐹 “ 𝐴)) = (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)))
29	20, 28	eqtrd 2798	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶onto𝑌) ∧ 𝐴 ⊆ 𝑌) → (◡𝐹 “ (𝑌 ∖ 𝐴)) = (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)))
30	29	eleq1d 2849	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶onto𝑌) ∧ 𝐴 ⊆ 𝑌) → ((◡𝐹 “ (𝑌 ∖ 𝐴)) ∈ 𝐽 ↔ (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)) ∈ 𝐽))
31		topontop 23286	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
32	31	ad2antrr 739	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶onto𝑌) ∧ 𝐴 ⊆ 𝑌) → 𝐽 ∈ Top)
33		cnvimass 6217	. . . . . . . . 9 ⊢ (◡𝐹 “ 𝐴) ⊆ dom 𝐹
34		fofn 6828	. . . . . . . . . . 11 ⊢ (𝐹:𝑋⟶onto𝑌 → 𝐹 Fn 𝑋)
35	34	fndmd 6674	. . . . . . . . . 10 ⊢ (𝐹:𝑋⟶onto𝑌 → dom 𝐹 = 𝑋)
36	35	ad2antlr 740	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶onto𝑌) ∧ 𝐴 ⊆ 𝑌) → dom 𝐹 = 𝑋)
37	33, 36	sseqtrid 3955	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶onto𝑌) ∧ 𝐴 ⊆ 𝑌) → (◡𝐹 “ 𝐴) ⊆ 𝑋)
38	37, 26	sseqtrd 3949	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶onto𝑌) ∧ 𝐴 ⊆ 𝑌) → (◡𝐹 “ 𝐴) ⊆ ∪ 𝐽)
39		eqid 2763	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
40	39	iscld2 23401	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (◡𝐹 “ 𝐴) ⊆ ∪ 𝐽) → ((◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)) ∈ 𝐽))
41	32, 38, 40	syl2anc 596	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶onto𝑌) ∧ 𝐴 ⊆ 𝑌) → ((◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)) ∈ 𝐽))
42	30, 41	bitr4d 285	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶onto𝑌) ∧ 𝐴 ⊆ 𝑌) → ((◡𝐹 “ (𝑌 ∖ 𝐴)) ∈ 𝐽 ↔ (◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽)))
43	15, 42	bitr3id 288	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶onto𝑌) ∧ 𝐴 ⊆ 𝑌) → (((𝑌 ∖ 𝐴) ⊆ 𝑌 ∧ (◡𝐹 “ (𝑌 ∖ 𝐴)) ∈ 𝐽) ↔ (◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽)))
44	13, 43	bitrd 282	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶onto𝑌) ∧ 𝐴 ⊆ 𝑌) → ((𝑌 ∖ 𝐴) ∈ (𝐽 qTop 𝐹) ↔ (◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽)))
45	44	pm5.32da 590	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶onto𝑌) → ((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⟶onto𝑌) → (𝐴 ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ (𝐴 ⊆ 𝑌 ∧ (◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900   ∖ cdif 4048  ∪ cuni 4872  ^◡ccnv 5657  dom cdm 5658   “ cima 5661  Fun wfun 6560  ⟶wf 6562  ⟶ontowfo 6564  ‘cfv 6566  (class class class)co 7451   qTop cqtop 17725  Topctop 23266  TopOnctopon 23283  Clsdccld 23389

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-qtop 17729  df-top 23267  df-topon 23284  df-cld 23392

This theorem is used by:  qtoprest  24091  kqcld  24109  qustgphaus  24497  qtopt1  34578