Theorem cnclsi 23159

Description: Property of the image of a closure. (Contributed by Mario Carneiro, 25-Aug-2015.)

Hypothesis

Ref	Expression
cnclsi.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
cnclsi	⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → (𝐹 “ ((cls‘𝐽)‘𝑆)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑆)))

Proof of Theorem cnclsi

Step	Hyp	Ref	Expression
1		cntop1 23127	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
2	1	adantr 480	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → 𝐽 ∈ Top)
3		cnvimass 6053	. . . . 5 ⊢ (◡𝐹 “ (𝐹 “ 𝑆)) ⊆ dom 𝐹
4		cnclsi.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
5		eqid 2729	. . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾
6	4, 5	cnf 23133	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶∪ 𝐾)
7	6	adantr 480	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → 𝐹:𝑋⟶∪ 𝐾)
8	3, 7	fssdm 6707	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ 𝑆)) ⊆ 𝑋)
9		simpr 484	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ 𝑋)
10	7	fdmd 6698	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → dom 𝐹 = 𝑋)
11	9, 10	sseqtrrd 3984	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ dom 𝐹)
12		sseqin2 4186	. . . . . 6 ⊢ (𝑆 ⊆ dom 𝐹 ↔ (dom 𝐹 ∩ 𝑆) = 𝑆)
13	11, 12	sylib 218	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → (dom 𝐹 ∩ 𝑆) = 𝑆)
14		dminss 6126	. . . . 5 ⊢ (dom 𝐹 ∩ 𝑆) ⊆ (◡𝐹 “ (𝐹 “ 𝑆))
15	13, 14	eqsstrrdi 3992	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ (◡𝐹 “ (𝐹 “ 𝑆)))
16	4	clsss 22941	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (◡𝐹 “ (𝐹 “ 𝑆)) ⊆ 𝑋 ∧ 𝑆 ⊆ (◡𝐹 “ (𝐹 “ 𝑆))) → ((cls‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘(◡𝐹 “ (𝐹 “ 𝑆))))
17	2, 8, 15, 16	syl3anc 1373	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘(◡𝐹 “ (𝐹 “ 𝑆))))
18		imassrn 6042	. . . . 5 ⊢ (𝐹 “ 𝑆) ⊆ ran 𝐹
19	7	frnd 6696	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → ran 𝐹 ⊆ ∪ 𝐾)
20	18, 19	sstrid 3958	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → (𝐹 “ 𝑆) ⊆ ∪ 𝐾)
21	5	cncls2i 23157	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐹 “ 𝑆) ⊆ ∪ 𝐾) → ((cls‘𝐽)‘(◡𝐹 “ (𝐹 “ 𝑆))) ⊆ (◡𝐹 “ ((cls‘𝐾)‘(𝐹 “ 𝑆))))
22	20, 21	syldan 591	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘(◡𝐹 “ (𝐹 “ 𝑆))) ⊆ (◡𝐹 “ ((cls‘𝐾)‘(𝐹 “ 𝑆))))
23	17, 22	sstrd 3957	. 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ (◡𝐹 “ ((cls‘𝐾)‘(𝐹 “ 𝑆))))
24	7	ffund 6692	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → Fun 𝐹)
25	4	clsss3 22946	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
26	1, 25	sylan 580	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
27	26, 10	sseqtrrd 3984	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ dom 𝐹)
28		funimass3 7026	. . 3 ⊢ ((Fun 𝐹 ∧ ((cls‘𝐽)‘𝑆) ⊆ dom 𝐹) → ((𝐹 “ ((cls‘𝐽)‘𝑆)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑆)) ↔ ((cls‘𝐽)‘𝑆) ⊆ (◡𝐹 “ ((cls‘𝐾)‘(𝐹 “ 𝑆)))))
29	24, 27, 28	syl2anc 584	. 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → ((𝐹 “ ((cls‘𝐽)‘𝑆)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑆)) ↔ ((cls‘𝐽)‘𝑆) ⊆ (◡𝐹 “ ((cls‘𝐾)‘(𝐹 “ 𝑆)))))
30	23, 29	mpbird 257	1 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → (𝐹 “ ((cls‘𝐽)‘𝑆)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑆)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∩ cin 3913   ⊆ wss 3914  ∪ cuni 4871  ^◡ccnv 5637  dom cdm 5638  ran crn 5639   “ cima 5641  Fun wfun 6505  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  Topctop 22780  clsccl 22905   Cn ccn 23111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-map 8801  df-top 22781  df-topon 22798  df-cld 22906  df-cls 22908  df-cn 23114

This theorem is referenced by:  cncls  23161  hmeocls  23655  clsnsg  23997