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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
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