Theorem qtopcld 23742

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 23733	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
2		topontop 22940	. . 3 ⊢ ((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) → (𝐽 qTop 𝐹) ∈ Top)
3		eqid 2740	. . . 4 ⊢ ∪ (𝐽 qTop 𝐹) = ∪ (𝐽 qTop 𝐹)
4	3	iscld 23056	. . 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 22941	. . . . 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 4041	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐴 ⊆ 𝑌 ↔ 𝐴 ⊆ ∪ (𝐽 qTop 𝐹)))
9	7	difeq1d 4148	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝑌 ∖ 𝐴) = (∪ (𝐽 qTop 𝐹) ∖ 𝐴))
10	9	eleq1d 2829	. . 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 23732	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → ((𝑌 ∖ 𝐴) ∈ (𝐽 qTop 𝐹) ↔ ((𝑌 ∖ 𝐴) ⊆ 𝑌 ∧ (◡𝐹 “ (𝑌 ∖ 𝐴)) ∈ 𝐽)))
13	12	adantr 480	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → ((𝑌 ∖ 𝐴) ∈ (𝐽 qTop 𝐹) ↔ ((𝑌 ∖ 𝐴) ⊆ 𝑌 ∧ (◡𝐹 “ (𝑌 ∖ 𝐴)) ∈ 𝐽)))
14		difss 4159	. . . . . 6 ⊢ (𝑌 ∖ 𝐴) ⊆ 𝑌
15	14	biantrur 530	. . . . 5 ⊢ ((◡𝐹 “ (𝑌 ∖ 𝐴)) ∈ 𝐽 ↔ ((𝑌 ∖ 𝐴) ⊆ 𝑌 ∧ (◡𝐹 “ (𝑌 ∖ 𝐴)) ∈ 𝐽))
16		fofun 6835	. . . . . . . . . 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 6645	. . . . . . . . 9 ⊢ (Fun 𝐹 → Fun ◡◡𝐹)
19		imadif 6662	. . . . . . . . 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 6834	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌)
22		fimacnv 6769	. . . . . . . . . . . 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 22941	. . . . . . . . . . 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 2780	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → (◡𝐹 “ 𝑌) = ∪ 𝐽)
28	27	difeq1d 4148	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → ((◡𝐹 “ 𝑌) ∖ (◡𝐹 “ 𝐴)) = (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)))
29	20, 28	eqtrd 2780	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → (◡𝐹 “ (𝑌 ∖ 𝐴)) = (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)))
30	29	eleq1d 2829	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → ((◡𝐹 “ (𝑌 ∖ 𝐴)) ∈ 𝐽 ↔ (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)) ∈ 𝐽))
31		topontop 22940	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
32	31	ad2antrr 725	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → 𝐽 ∈ Top)
33		cnvimass 6111	. . . . . . . . 9 ⊢ (◡𝐹 “ 𝐴) ⊆ dom 𝐹
34		fofn 6836	. . . . . . . . . . 11 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹 Fn 𝑋)
35	34	fndmd 6684	. . . . . . . . . 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 4061	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → (◡𝐹 “ 𝐴) ⊆ 𝑋)
38	37, 26	sseqtrd 4049	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → (◡𝐹 “ 𝐴) ⊆ ∪ 𝐽)
39		eqid 2740	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
40	39	iscld2 23057	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (◡𝐹 “ 𝐴) ⊆ ∪ 𝐽) → ((◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)) ∈ 𝐽))
41	32, 38, 40	syl2anc 583	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → ((◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ (◡𝐹 “ 𝐴)) ∈ 𝐽))
42	30, 41	bitr4d 282	. . . . 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 279	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) ∧ 𝐴 ⊆ 𝑌) → ((𝑌 ∖ 𝐴) ∈ (𝐽 qTop 𝐹) ↔ (◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽)))
45	44	pm5.32da 578	. 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 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ∖ cdif 3973   ⊆ wss 3976  ∪ cuni 4931  ^◡ccnv 5699  dom cdm 5700   “ cima 5703  Fun wfun 6567  ⟶wf 6569  –onto→wfo 6571  ‘cfv 6573  (class class class)co 7448   qTop cqtop 17563  Topctop 22920  TopOnctopon 22937  Clsdccld 23045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-qtop 17567  df-top 22921  df-topon 22938  df-cld 23048

This theorem is referenced by:  qtoprest  23746  kqcld  23764  qustgphaus  24152  qtopt1  33781