Theorem hmeocls 23655

Description: Homeomorphisms preserve closures. (Contributed by Mario Carneiro, 25-Aug-2015.)

Hypothesis

Ref	Expression
hmeoopn.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
hmeocls	⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐾)‘(𝐹 “ 𝐴)) = (𝐹 “ ((cls‘𝐽)‘𝐴)))

Proof of Theorem hmeocls

Step	Hyp	Ref	Expression
1		hmeocnvcn 23648	. . . 4 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾 Cn 𝐽))
2		hmeoopn.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
3	2	cncls2i 23157	. . . 4 ⊢ ((◡𝐹 ∈ (𝐾 Cn 𝐽) ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐾)‘(◡◡𝐹 “ 𝐴)) ⊆ (◡◡𝐹 “ ((cls‘𝐽)‘𝐴)))
4	1, 3	sylan 580	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐾)‘(◡◡𝐹 “ 𝐴)) ⊆ (◡◡𝐹 “ ((cls‘𝐽)‘𝐴)))
5		imacnvcnv 6179	. . . 4 ⊢ (◡◡𝐹 “ 𝐴) = (𝐹 “ 𝐴)
6	5	fveq2i 6861	. . 3 ⊢ ((cls‘𝐾)‘(◡◡𝐹 “ 𝐴)) = ((cls‘𝐾)‘(𝐹 “ 𝐴))
7		imacnvcnv 6179	. . 3 ⊢ (◡◡𝐹 “ ((cls‘𝐽)‘𝐴)) = (𝐹 “ ((cls‘𝐽)‘𝐴))
8	4, 6, 7	3sstr3g 3999	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐾)‘(𝐹 “ 𝐴)) ⊆ (𝐹 “ ((cls‘𝐽)‘𝐴)))
9		hmeocn 23647	. . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
10	2	cnclsi 23159	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐹 “ ((cls‘𝐽)‘𝐴)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝐴)))
11	9, 10	sylan 580	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐹 “ ((cls‘𝐽)‘𝐴)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝐴)))
12	8, 11	eqssd 3964	1 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐾)‘(𝐹 “ 𝐴)) = (𝐹 “ ((cls‘𝐽)‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ⊆ wss 3914  ∪ cuni 4871  ^◡ccnv 5637   “ cima 5641  ‘cfv 6511  (class class class)co 7387  clsccl 22905   Cn ccn 23111  Homeochmeo 23640

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  df-hmeo 23642

This theorem is referenced by:  reghmph  23680  nrmhmph  23681  snclseqg  24003