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Theorem cnpf2 21227
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 2748 . . . 4 𝐽 = 𝐽
2 eqid 2748 . . . 4 𝐾 = 𝐾
31, 2cnpf 21224 . . 3 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐹: 𝐽 𝐾)
4 toponuni 20892 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
54feq2d 6180 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (𝐹:𝑋𝑌𝐹: 𝐽𝑌))
6 toponuni 20892 . . . . 5 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
76feq3d 6181 . . . 4 (𝐾 ∈ (TopOn‘𝑌) → (𝐹: 𝐽𝑌𝐹: 𝐽 𝐾))
85, 7sylan9bb 738 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹:𝑋𝑌𝐹: 𝐽 𝐾))
93, 8syl5ibr 236 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐹:𝑋𝑌))
1093impia 1109 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 1072  wcel 2127   cuni 4576  wf 6033  cfv 6037  (class class class)co 6801  TopOnctopon 20888   CnP ccnp 21202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-fv 6045  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-1st 7321  df-2nd 7322  df-map 8013  df-top 20872  df-topon 20889  df-cnp 21205
This theorem is referenced by:  iscnp4  21240  1stccnp  21438  txcnp  21596  ptcnplem  21597  ptcnp  21598  cnpflf2  21976  cnpflf  21977  flfcnp  21980  flfcnp2  21983  cnpfcf  22017  ghmcnp  22090  metcnpi3  22523  limcvallem  23805  cnplimc  23821  limccnp  23825  limccnp2  23826  ftc1lem3  23971
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