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Theorem hmeocls 22311
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 22304 . . . 4 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾 Cn 𝐽))
2 hmeoopn.1 . . . . 5 𝑋 = 𝐽
32cncls2i 21813 . . . 4 ((𝐹 ∈ (𝐾 Cn 𝐽) ∧ 𝐴𝑋) → ((cls‘𝐾)‘(𝐹𝐴)) ⊆ (𝐹 “ ((cls‘𝐽)‘𝐴)))
41, 3sylan 580 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((cls‘𝐾)‘(𝐹𝐴)) ⊆ (𝐹 “ ((cls‘𝐽)‘𝐴)))
5 imacnvcnv 6062 . . . 4 (𝐹𝐴) = (𝐹𝐴)
65fveq2i 6672 . . 3 ((cls‘𝐾)‘(𝐹𝐴)) = ((cls‘𝐾)‘(𝐹𝐴))
7 imacnvcnv 6062 . . 3 (𝐹 “ ((cls‘𝐽)‘𝐴)) = (𝐹 “ ((cls‘𝐽)‘𝐴))
84, 6, 73sstr3g 4015 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((cls‘𝐾)‘(𝐹𝐴)) ⊆ (𝐹 “ ((cls‘𝐽)‘𝐴)))
9 hmeocn 22303 . . 3 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
102cnclsi 21815 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) → (𝐹 “ ((cls‘𝐽)‘𝐴)) ⊆ ((cls‘𝐾)‘(𝐹𝐴)))
119, 10sylan 580 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐹 “ ((cls‘𝐽)‘𝐴)) ⊆ ((cls‘𝐾)‘(𝐹𝐴)))
128, 11eqssd 3988 1 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((cls‘𝐾)‘(𝐹𝐴)) = (𝐹 “ ((cls‘𝐽)‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  wss 3940   cuni 4837  ccnv 5553  cima 5557  cfv 6354  (class class class)co 7150  clsccl 21561   Cn ccn 21767  Homeochmeo 22296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-iin 4920  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7153  df-oprab 7154  df-mpo 7155  df-map 8403  df-top 21437  df-topon 21454  df-cld 21562  df-cls 21564  df-cn 21770  df-hmeo 22298
This theorem is referenced by:  reghmph  22336  nrmhmph  22337  snclseqg  22658
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