MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hmeocls Structured version   Visualization version   GIF version

Theorem hmeocls 21565
Description: Homeomorphisms preserve closures. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
hmeoopn.1 𝑋 = 𝐽
Assertion
Ref Expression
hmeocls ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((cls‘𝐾)‘(𝐹𝐴)) = (𝐹 “ ((cls‘𝐽)‘𝐴)))

Proof of Theorem hmeocls
StepHypRef Expression
1 hmeocnvcn 21558 . . . 4 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾 Cn 𝐽))
2 hmeoopn.1 . . . . 5 𝑋 = 𝐽
32cncls2i 21068 . . . 4 ((𝐹 ∈ (𝐾 Cn 𝐽) ∧ 𝐴𝑋) → ((cls‘𝐾)‘(𝐹𝐴)) ⊆ (𝐹 “ ((cls‘𝐽)‘𝐴)))
41, 3sylan 488 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((cls‘𝐾)‘(𝐹𝐴)) ⊆ (𝐹 “ ((cls‘𝐽)‘𝐴)))
5 imacnvcnv 5597 . . . 4 (𝐹𝐴) = (𝐹𝐴)
65fveq2i 6192 . . 3 ((cls‘𝐾)‘(𝐹𝐴)) = ((cls‘𝐾)‘(𝐹𝐴))
7 imacnvcnv 5597 . . 3 (𝐹 “ ((cls‘𝐽)‘𝐴)) = (𝐹 “ ((cls‘𝐽)‘𝐴))
84, 6, 73sstr3g 3643 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((cls‘𝐾)‘(𝐹𝐴)) ⊆ (𝐹 “ ((cls‘𝐽)‘𝐴)))
9 hmeocn 21557 . . 3 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
102cnclsi 21070 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) → (𝐹 “ ((cls‘𝐽)‘𝐴)) ⊆ ((cls‘𝐾)‘(𝐹𝐴)))
119, 10sylan 488 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐹 “ ((cls‘𝐽)‘𝐴)) ⊆ ((cls‘𝐾)‘(𝐹𝐴)))
128, 11eqssd 3618 1 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((cls‘𝐾)‘(𝐹𝐴)) = (𝐹 “ ((cls‘𝐽)‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1482  wcel 1989  wss 3572   cuni 4434  ccnv 5111  cima 5115  cfv 5886  (class class class)co 6647  clsccl 20816   Cn ccn 21022  Homeochmeo 21550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-iin 4521  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-map 7856  df-top 20693  df-topon 20710  df-cld 20817  df-cls 20819  df-cn 21025  df-hmeo 21552
This theorem is referenced by:  reghmph  21590  nrmhmph  21591  snclseqg  21913
  Copyright terms: Public domain W3C validator