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Theorem cnpf2 23274
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 2735 . . . 4 𝐽 = 𝐽
2 eqid 2735 . . . 4 𝐾 = 𝐾
31, 2cnpf 23271 . . 3 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐹: 𝐽 𝐾)
4 toponuni 22936 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
54feq2d 6723 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (𝐹:𝑋𝑌𝐹: 𝐽𝑌))
6 toponuni 22936 . . . . 5 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
76feq3d 6724 . . . 4 (𝐾 ∈ (TopOn‘𝑌) → (𝐹: 𝐽𝑌𝐹: 𝐽 𝐾))
85, 7sylan9bb 509 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹:𝑋𝑌𝐹: 𝐽 𝐾))
93, 8imbitrrid 246 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐹:𝑋𝑌))
1093impia 1116 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 395  w3a 1086  wcel 2106   cuni 4912  wf 6559  cfv 6563  (class class class)co 7431  TopOnctopon 22932   CnP ccnp 23249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-top 22916  df-topon 22933  df-cnp 23252
This theorem is referenced by:  iscnp4  23287  1stccnp  23486  txcnp  23644  ptcnplem  23645  ptcnp  23646  cnpflf2  24024  cnpflf  24025  flfcnp  24028  flfcnp2  24031  cnpfcf  24065  ghmcnp  24139  metcnpi3  24575  limcvallem  25921  cnplimc  25937  limccnp  25941  limccnp2  25942  ftc1lem3  26094
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