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Theorem cnpf 21841
 Description: A continuous function at point 𝑃 is a mapping. (Contributed by FL, 17-Nov-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
Hypotheses
Ref Expression
iscnp2.1 𝑋 = 𝐽
iscnp2.2 𝑌 = 𝐾
Assertion
Ref Expression
cnpf (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐹:𝑋𝑌)

Proof of Theorem cnpf
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iscnp2.1 . . . 4 𝑋 = 𝐽
2 iscnp2.2 . . . 4 𝑌 = 𝐾
31, 2iscnp2 21833 . . 3 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃𝑋) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))))
43simprbi 500 . 2 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦))))
54simpld 498 1 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐹:𝑋𝑌)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115  ∀wral 3132  ∃wrex 3133   ⊆ wss 3918  ∪ cuni 4819   “ cima 5539  ⟶wf 6332  ‘cfv 6336  (class class class)co 7138  Topctop 21487   CnP ccnp 21819 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-fv 6344  df-ov 7141  df-oprab 7142  df-mpo 7143  df-1st 7672  df-2nd 7673  df-map 8391  df-top 21488  df-topon 21505  df-cnp 21822 This theorem is referenced by:  cnpcl  21842  cnpf2  21844  cnpco  21861  cncnp2  21875  cnpresti  21882  lmcnp  21898  txcnpi  22202  cnpflfi  22593  cnpfcfi  22634
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