Theorem cnclsi 22776

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 22744	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
2	1	adantr 482	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → 𝐽 ∈ Top)
3		cnvimass 6081	. . . . 5 ⊢ (◡𝐹 “ (𝐹 “ 𝑆)) ⊆ dom 𝐹
4		cnclsi.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
5		eqid 2733	. . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾
6	4, 5	cnf 22750	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶∪ 𝐾)
7	6	adantr 482	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → 𝐹:𝑋⟶∪ 𝐾)
8	3, 7	fssdm 6738	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ 𝑆)) ⊆ 𝑋)
9		simpr 486	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ 𝑋)
10	7	fdmd 6729	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → dom 𝐹 = 𝑋)
11	9, 10	sseqtrrd 4024	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ dom 𝐹)
12		sseqin2 4216	. . . . . 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 6153	. . . . 5 ⊢ (dom 𝐹 ∩ 𝑆) ⊆ (◡𝐹 “ (𝐹 “ 𝑆))
15	13, 14	eqsstrrdi 4038	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ (◡𝐹 “ (𝐹 “ 𝑆)))
16	4	clsss 22558	. . . 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 1372	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘(◡𝐹 “ (𝐹 “ 𝑆))))
18		imassrn 6071	. . . . 5 ⊢ (𝐹 “ 𝑆) ⊆ ran 𝐹
19	7	frnd 6726	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → ran 𝐹 ⊆ ∪ 𝐾)
20	18, 19	sstrid 3994	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → (𝐹 “ 𝑆) ⊆ ∪ 𝐾)
21	5	cncls2i 22774	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐹 “ 𝑆) ⊆ ∪ 𝐾) → ((cls‘𝐽)‘(◡𝐹 “ (𝐹 “ 𝑆))) ⊆ (◡𝐹 “ ((cls‘𝐾)‘(𝐹 “ 𝑆))))
22	20, 21	syldan 592	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘(◡𝐹 “ (𝐹 “ 𝑆))) ⊆ (◡𝐹 “ ((cls‘𝐾)‘(𝐹 “ 𝑆))))
23	17, 22	sstrd 3993	. 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ (◡𝐹 “ ((cls‘𝐾)‘(𝐹 “ 𝑆))))
24	7	ffund 6722	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → Fun 𝐹)
25	4	clsss3 22563	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
26	1, 25	sylan 581	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
27	26, 10	sseqtrrd 4024	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ dom 𝐹)
28		funimass3 7056	. . 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 585	. 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 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   ∩ cin 3948   ⊆ wss 3949  ∪ cuni 4909  ^◡ccnv 5676  dom cdm 5677  ran crn 5678   “ cima 5680  Fun wfun 6538  ⟶wf 6540  ‘cfv 6544  (class class class)co 7409  Topctop 22395  clsccl 22522   Cn ccn 22728

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-map 8822  df-top 22396  df-topon 22413  df-cld 22523  df-cls 22525  df-cn 22731

This theorem is referenced by:  cncls  22778  hmeocls  23272  clsnsg  23614