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Theorem cnpf2 23259
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 2736 . . . 4 𝐽 = 𝐽
2 eqid 2736 . . . 4 𝐾 = 𝐾
31, 2cnpf 23256 . . 3 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐹: 𝐽 𝐾)
4 toponuni 22921 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
54feq2d 6721 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (𝐹:𝑋𝑌𝐹: 𝐽𝑌))
6 toponuni 22921 . . . . 5 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
76feq3d 6722 . . . 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 1117 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 2107   cuni 4906  wf 6556  cfv 6560  (class class class)co 7432  TopOnctopon 22917   CnP ccnp 23234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-map 8869  df-top 22901  df-topon 22918  df-cnp 23237
This theorem is referenced by:  iscnp4  23272  1stccnp  23471  txcnp  23629  ptcnplem  23630  ptcnp  23631  cnpflf2  24009  cnpflf  24010  flfcnp  24013  flfcnp2  24016  cnpfcf  24050  ghmcnp  24124  metcnpi3  24560  limcvallem  25907  cnplimc  25923  limccnp  25927  limccnp2  25928  ftc1lem3  26080
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