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Theorem cnptop2 21854
Description: Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
cnptop2 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐾 ∈ Top)

Proof of Theorem cnptop2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2824 . . . 4 𝐽 = 𝐽
2 eqid 2824 . . . 4 𝐾 = 𝐾
31, 2iscnp2 21850 . . 3 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃 𝐽) ∧ (𝐹: 𝐽 𝐾 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))))
43simplbi 500 . 2 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃 𝐽))
54simp2d 1139 1 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐾 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1083  wcel 2113  wral 3141  wrex 3142  wss 3939   cuni 4841  cima 5561  wf 6354  cfv 6358  (class class class)co 7159  Topctop 21504   CnP ccnp 21836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-1st 7692  df-2nd 7693  df-map 8411  df-top 21505  df-topon 21522  df-cnp 21839
This theorem is referenced by:  cnpco  21878  cncnp2  21892  cnpresti  21899  cnprest  21900  lmcnp  21915  cnpflfi  22610  flfcnp  22615  flfcnp2  22618
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