Theorem hmeocls 23758

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 23751	. . . 4 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾 Cn 𝐽))
2		hmeoopn.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
3	2	cncls2i 23260	. . . 4 ⊢ ((◡𝐹 ∈ (𝐾 Cn 𝐽) ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐾)‘(◡◡𝐹 “ 𝐴)) ⊆ (◡◡𝐹 “ ((cls‘𝐽)‘𝐴)))
4	1, 3	sylan 586	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐾)‘(◡◡𝐹 “ 𝐴)) ⊆ (◡◡𝐹 “ ((cls‘𝐽)‘𝐴)))
5		imacnvcnv 6164	. . . 4 ⊢ (◡◡𝐹 “ 𝐴) = (𝐹 “ 𝐴)
6	5	fveq2i 6837	. . 3 ⊢ ((cls‘𝐾)‘(◡◡𝐹 “ 𝐴)) = ((cls‘𝐾)‘(𝐹 “ 𝐴))
7		imacnvcnv 6164	. . 3 ⊢ (◡◡𝐹 “ ((cls‘𝐽)‘𝐴)) = (𝐹 “ ((cls‘𝐽)‘𝐴))
8	4, 6, 7	3sstr3g 3974	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐾)‘(𝐹 “ 𝐴)) ⊆ (𝐹 “ ((cls‘𝐽)‘𝐴)))
9		hmeocn 23750	. . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
10	2	cnclsi 23262	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐹 “ ((cls‘𝐽)‘𝐴)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝐴)))
11	9, 10	sylan 586	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐹 “ ((cls‘𝐽)‘𝐴)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝐴)))
12	8, 11	eqssd 3939	1 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐾)‘(𝐹 “ 𝐴)) = (𝐹 “ ((cls‘𝐽)‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ⊆ wss 3890  ∪ cuni 4845  ^◡ccnv 5624   “ cima 5628  ‘cfv 6492  (class class class)co 7363  clsccl 23008   Cn ccn 23214  Homeochmeo 23743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-iin 4931  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-map 8772  df-top 22884  df-topon 22901  df-cld 23009  df-cls 23011  df-cn 23217  df-hmeo 23745

This theorem is referenced by:  reghmph  23783  nrmhmph  23784  snclseqg  24106