Theorem cnclsi 23537

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 23505	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
2	1	adantr 486	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → 𝐽 ∈ Top)
3		cnvimass 6073	. . . . 5 ⊢ (◡𝐹 “ (𝐹 “ 𝑆)) ⊆ dom 𝐹
4		cnclsi.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
5		eqid 2760	. . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾
6	4, 5	cnf 23511	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶∪ 𝐾)
7	6	adantr 486	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → 𝐹:𝑋⟶∪ 𝐾)
8	3, 7	fssdm 6718	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ 𝑆)) ⊆ 𝑋)
9		simpr 490	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ 𝑋)
10	7	fdmd 6709	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → dom 𝐹 = 𝑋)
11	9, 10	sseqtrrd 3968	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ dom 𝐹)
12		sseqin2 4169	. . . . . 6 ⊢ (𝑆 ⊆ dom 𝐹 ↔ (dom 𝐹 ∩ 𝑆) = 𝑆)
13	11, 12	sylib 221	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → (dom 𝐹 ∩ 𝑆) = 𝑆)
14		dminss 6139	. . . . 5 ⊢ (dom 𝐹 ∩ 𝑆) ⊆ (◡𝐹 “ (𝐹 “ 𝑆))
15	13, 14	eqsstrrdi 3976	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ (◡𝐹 “ (𝐹 “ 𝑆)))
16	4	clsss 23319	. . . 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 1398	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘(◡𝐹 “ (𝐹 “ 𝑆))))
18		imassrn 6062	. . . . 5 ⊢ (𝐹 “ 𝑆) ⊆ ran 𝐹
19	7	frnd 6707	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → ran 𝐹 ⊆ ∪ 𝐾)
20	18, 19	sstrid 3942	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → (𝐹 “ 𝑆) ⊆ ∪ 𝐾)
21	5	cncls2i 23535	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐹 “ 𝑆) ⊆ ∪ 𝐾) → ((cls‘𝐽)‘(◡𝐹 “ (𝐹 “ 𝑆))) ⊆ (◡𝐹 “ ((cls‘𝐾)‘(𝐹 “ 𝑆))))
22	20, 21	syldan 603	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘(◡𝐹 “ (𝐹 “ 𝑆))) ⊆ (◡𝐹 “ ((cls‘𝐾)‘(𝐹 “ 𝑆))))
23	17, 22	sstrd 3941	. 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ (◡𝐹 “ ((cls‘𝐾)‘(𝐹 “ 𝑆))))
24	7	ffund 6703	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → Fun 𝐹)
25	4	clsss3 23324	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
26	1, 25	sylan 592	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
27	26, 10	sseqtrrd 3968	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ dom 𝐹)
28		funimass3 7042	. . 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 596	. 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 260	1 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → (𝐹 “ ((cls‘𝐽)‘𝑆)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑆)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ∩ cin 3898   ⊆ wss 3899  ∪ cuni 4867  ^◡ccnv 5647  dom cdm 5648  ran crn 5649   “ cima 5651  Fun wfun 6522  ⟶wf 6524  ‘cfv 6528  (class class class)co 7409  Topctop 23158  clsccl 23283   Cn ccn 23489

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7735

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6484  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-ov 7412  df-oprab 7413  df-mpo 7414  df-map 8828  df-top 23159  df-topon 23176  df-cld 23284  df-cls 23286  df-cn 23492

This theorem is used by:  cncls  23539  hmeocls  24034  clsnsg  24376