Theorem hmeocls 21571

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

Hypothesis

Ref	Expression
hmeoopn.1	|- X = U. J

Assertion

Ref	Expression
hmeocls	|- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( cls ` K ) ` ( F " A ) ) = ( F " ( ( cls ` J ) ` A ) ) )

Proof of Theorem hmeocls

Step	Hyp	Ref	Expression
1		hmeocnvcn 21564	. . . 4 |- ( F e. ( J Homeo K ) -> `' F e. ( K Cn J ) )
2		hmeoopn.1	. . . . 5 |- X = U. J
3	2	cncls2i 21074	. . . 4 |- ( ( `' F e. ( K Cn J ) /\ A C_ X ) -> ( ( cls ` K ) ` ( `' `' F " A ) ) C_ ( `' `' F " ( ( cls ` J ) ` A ) ) )
4	1, 3	sylan 488	. . 3 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( cls ` K ) ` ( `' `' F " A ) ) C_ ( `' `' F " ( ( cls ` J ) ` A ) ) )
5		imacnvcnv 5599	. . . 4 |- ( `' `' F " A ) = ( F " A )
6	5	fveq2i 6194	. . 3 |- ( ( cls ` K ) ` ( `' `' F " A ) ) = ( ( cls ` K ) ` ( F " A ) )
7		imacnvcnv 5599	. . 3 |- ( `' `' F " ( ( cls ` J ) ` A ) ) = ( F " ( ( cls ` J ) ` A ) )
8	4, 6, 7	3sstr3g 3645	. 2 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( cls ` K ) ` ( F " A ) ) C_ ( F " ( ( cls ` J ) ` A ) ) )
9		hmeocn 21563	. . 3 |- ( F e. ( J Homeo K ) -> F e. ( J Cn K ) )
10	2	cnclsi 21076	. . 3 |- ( ( F e. ( J Cn K ) /\ A C_ X ) -> ( F " ( ( cls ` J ) ` A ) ) C_ ( ( cls ` K ) ` ( F " A ) ) )
11	9, 10	sylan 488	. 2 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( F " ( ( cls ` J ) ` A ) ) C_ ( ( cls ` K ) ` ( F " A ) ) )
12	8, 11	eqssd 3620	1 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( cls ` K ) ` ( F " A ) ) = ( F " ( ( cls ` J ) ` A ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   U.cuni 4436   `'ccnv 5113   "cima 5117   ` cfv 5888  (class class class)co 6650   clsccl 20822   Cn ccn 21028   Homeochmeo 21556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-top 20699  df-topon 20716  df-cld 20823  df-cls 20825  df-cn 21031  df-hmeo 21558

This theorem is referenced by:  reghmph  21596  nrmhmph  21597  snclseqg  21919