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Theorem cnpf2 20264
Description: A continuous function at point  P is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
cnpf2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  (
( J  CnP  K
) `  P )
)  ->  F : X
--> Y )

Proof of Theorem cnpf2
StepHypRef Expression
1 eqid 2422 . . . 4  |-  U. J  =  U. J
2 eqid 2422 . . . 4  |-  U. K  =  U. K
31, 2cnpf 20261 . . 3  |-  ( F  e.  ( ( J  CnP  K ) `  P )  ->  F : U. J --> U. K
)
4 toponuni 19940 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
54feq2d 5733 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  ( F : X --> Y  <->  F : U. J --> Y ) )
6 toponuni 19940 . . . . 5  |-  ( K  e.  (TopOn `  Y
)  ->  Y  =  U. K )
76feq3d 5734 . . . 4  |-  ( K  e.  (TopOn `  Y
)  ->  ( F : U. J --> Y  <->  F : U. J --> U. K ) )
85, 7sylan9bb 704 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( F : X --> Y  <->  F : U. J --> U. K ) )
93, 8syl5ibr 224 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( F  e.  ( ( J  CnP  K ) `  P )  ->  F : X --> Y ) )
1093impia 1202 1  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  (
( J  CnP  K
) `  P )
)  ->  F : X
--> Y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    e. wcel 1872   U.cuni 4219   -->wf 5597   ` cfv 5601  (class class class)co 6305  TopOnctopon 19916    CnP ccnp 20239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-map 7485  df-top 19919  df-topon 19921  df-cnp 20242
This theorem is referenced by:  iscnp4  20277  1stccnp  20475  txcnp  20633  ptcnplem  20634  ptcnp  20635  cnpflf2  21013  cnpflf  21014  flfcnp  21017  flfcnp2  21020  cnpfcf  21054  ghmcnp  21127  metcnpi3  21559  limcvallem  22824  cnplimc  22840  limccnp  22844  limccnp2  22845  ftc1lem3  22988
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