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Theorem cnpf2 20864
Description: A continuous function at point 𝑃 is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
cnpf2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:𝑋𝑌)

Proof of Theorem cnpf2
StepHypRef Expression
1 eqid 2610 . . . 4 𝐽 = 𝐽
2 eqid 2610 . . . 4 𝐾 = 𝐾
31, 2cnpf 20861 . . 3 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐹: 𝐽 𝐾)
4 toponuni 20542 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
54feq2d 5944 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (𝐹:𝑋𝑌𝐹: 𝐽𝑌))
6 toponuni 20542 . . . . 5 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
76feq3d 5945 . . . 4 (𝐾 ∈ (TopOn‘𝑌) → (𝐹: 𝐽𝑌𝐹: 𝐽 𝐾))
85, 7sylan9bb 732 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹:𝑋𝑌𝐹: 𝐽 𝐾))
93, 8syl5ibr 235 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐹:𝑋𝑌))
1093impia 1253 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:𝑋𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031  wcel 1977   cuni 4372  wf 5800  cfv 5804  (class class class)co 6549  TopOnctopon 20518   CnP ccnp 20839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-top 20521  df-topon 20523  df-cnp 20842
This theorem is referenced by:  iscnp4  20877  1stccnp  21075  txcnp  21233  ptcnplem  21234  ptcnp  21235  cnpflf2  21614  cnpflf  21615  flfcnp  21618  flfcnp2  21621  cnpfcf  21655  ghmcnp  21728  metcnpi3  22161  limcvallem  23441  cnplimc  23457  limccnp  23461  limccnp2  23462  ftc1lem3  23605
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