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Theorem cnprcl2 21426
Description: Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
cnprcl2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃𝑋)

Proof of Theorem cnprcl2
StepHypRef Expression
1 eqid 2825 . . . 4 𝐽 = 𝐽
21cnprcl 21420 . . 3 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝑃 𝐽)
32adantl 475 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃 𝐽)
4 toponuni 21089 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
54adantr 474 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑋 = 𝐽)
63, 5eleqtrrd 2909 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1658  wcel 2166   cuni 4658  cfv 6123  (class class class)co 6905  TopOnctopon 21085   CnP ccnp 21400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-1st 7428  df-2nd 7429  df-map 8124  df-top 21069  df-topon 21086  df-cnp 21403
This theorem is referenced by: (None)
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