Theorem cnclsi 23264

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 23232	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
2	1	adantr 479	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → 𝐽 ∈ Top)
3		cnvimass 6083	. . . . 5 ⊢ (◡𝐹 “ (𝐹 “ 𝑆)) ⊆ dom 𝐹
4		cnclsi.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
5		eqid 2726	. . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾
6	4, 5	cnf 23238	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶∪ 𝐾)
7	6	adantr 479	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → 𝐹:𝑋⟶∪ 𝐾)
8	3, 7	fssdm 6739	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ 𝑆)) ⊆ 𝑋)
9		simpr 483	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ 𝑋)
10	7	fdmd 6730	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → dom 𝐹 = 𝑋)
11	9, 10	sseqtrrd 4020	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ dom 𝐹)
12		sseqin2 4213	. . . . . 6 ⊢ (𝑆 ⊆ dom 𝐹 ↔ (dom 𝐹 ∩ 𝑆) = 𝑆)
13	11, 12	sylib 217	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → (dom 𝐹 ∩ 𝑆) = 𝑆)
14		dminss 6156	. . . . 5 ⊢ (dom 𝐹 ∩ 𝑆) ⊆ (◡𝐹 “ (𝐹 “ 𝑆))
15	13, 14	eqsstrrdi 4034	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ (◡𝐹 “ (𝐹 “ 𝑆)))
16	4	clsss 23046	. . . 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 1368	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘(◡𝐹 “ (𝐹 “ 𝑆))))
18		imassrn 6072	. . . . 5 ⊢ (𝐹 “ 𝑆) ⊆ ran 𝐹
19	7	frnd 6728	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → ran 𝐹 ⊆ ∪ 𝐾)
20	18, 19	sstrid 3990	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → (𝐹 “ 𝑆) ⊆ ∪ 𝐾)
21	5	cncls2i 23262	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐹 “ 𝑆) ⊆ ∪ 𝐾) → ((cls‘𝐽)‘(◡𝐹 “ (𝐹 “ 𝑆))) ⊆ (◡𝐹 “ ((cls‘𝐾)‘(𝐹 “ 𝑆))))
22	20, 21	syldan 589	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘(◡𝐹 “ (𝐹 “ 𝑆))) ⊆ (◡𝐹 “ ((cls‘𝐾)‘(𝐹 “ 𝑆))))
23	17, 22	sstrd 3989	. 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ (◡𝐹 “ ((cls‘𝐾)‘(𝐹 “ 𝑆))))
24	7	ffund 6724	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → Fun 𝐹)
25	4	clsss3 23051	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
26	1, 25	sylan 578	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
27	26, 10	sseqtrrd 4020	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ dom 𝐹)
28		funimass3 7059	. . 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 582	. 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 256	1 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → (𝐹 “ ((cls‘𝐽)‘𝑆)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑆)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1534   ∈ wcel 2099   ∩ cin 3945   ⊆ wss 3946  ∪ cuni 4905  ^◡ccnv 5673  dom cdm 5674  ran crn 5675   “ cima 5677  Fun wfun 6540  ⟶wf 6542  ‘cfv 6546  (class class class)co 7416  Topctop 22883  clsccl 23010   Cn ccn 23216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-int 4947  df-iun 4995  df-iin 4996  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7419  df-oprab 7420  df-mpo 7421  df-map 8849  df-top 22884  df-topon 22901  df-cld 23011  df-cls 23013  df-cn 23219

This theorem is referenced by:  cncls  23266  hmeocls  23760  clsnsg  24102