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Theorem cnptop1 22144
Description: Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
cnptop1 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐽 ∈ Top)

Proof of Theorem cnptop1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . . 4 𝐽 = 𝐽
2 eqid 2737 . . . 4 𝐾 = 𝐾
31, 2iscnp2 22141 . . 3 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃 𝐽) ∧ (𝐹: 𝐽 𝐾 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))))
43simplbi 501 . 2 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃 𝐽))
54simp1d 1144 1 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1089  wcel 2110  wral 3061  wrex 3062  wss 3871   cuni 4824  cima 5559  wf 6381  cfv 6385  (class class class)co 7218  Topctop 21795   CnP ccnp 22127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5197  ax-nul 5204  ax-pow 5263  ax-pr 5327  ax-un 7528
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3415  df-sbc 3700  df-csb 3817  df-dif 3874  df-un 3876  df-in 3878  df-ss 3888  df-nul 4243  df-if 4445  df-pw 4520  df-sn 4547  df-pr 4549  df-op 4553  df-uni 4825  df-iun 4911  df-br 5059  df-opab 5121  df-mpt 5141  df-id 5460  df-xp 5562  df-rel 5563  df-cnv 5564  df-co 5565  df-dm 5566  df-rn 5567  df-res 5568  df-ima 5569  df-iota 6343  df-fun 6387  df-fn 6388  df-f 6389  df-fv 6393  df-ov 7221  df-oprab 7222  df-mpo 7223  df-1st 7766  df-2nd 7767  df-map 8515  df-top 21796  df-topon 21813  df-cnp 22130
This theorem is referenced by:  cnpco  22169  cncnp2  22183  cnpresti  22190  cnprest2  22192  lmcnp  22206
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