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Theorem cnextrel 21127
Description: In the general case, a continuous extension is a relation. (Contributed by Thierry Arnoux, 20-Dec-2017.)
Hypotheses
Ref Expression
cnextfrel.1  |-  C  = 
U. J
cnextfrel.2  |-  B  = 
U. K
Assertion
Ref Expression
cnextrel  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  C
) )  ->  Rel  ( ( JCnExt K
) `  F )
)

Proof of Theorem cnextrel
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 relxp 4961 . . . 4  |-  Rel  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )
21rgenw 2761 . . 3  |-  A. x  e.  ( ( cls `  J
) `  A ) Rel  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )
3 reliun 4973 . . 3  |-  ( Rel  U_ x  e.  (
( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  <->  A. x  e.  ( ( cls `  J
) `  A ) Rel  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )
42, 3mpbir 214 . 2  |-  Rel  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )
5 cnextfrel.1 . . . 4  |-  C  = 
U. J
6 cnextfrel.2 . . . 4  |-  B  = 
U. K
75, 6cnextfval 21126 . . 3  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  C
) )  ->  (
( JCnExt K ) `
 F )  = 
U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )
87releqd 4938 . 2  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  C
) )  ->  ( Rel  ( ( JCnExt K
) `  F )  <->  Rel  U_ x  e.  (
( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) ) )
94, 8mpbiri 241 1  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  C
) )  ->  Rel  ( ( JCnExt K
) `  F )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    = wceq 1455    e. wcel 1898   A.wral 2749    C_ wss 3416   {csn 3980   U.cuni 4212   U_ciun 4292    X. cxp 4851   Rel wrel 4858   -->wf 5597   ` cfv 5601  (class class class)co 6315   ↾t crest 15368   Topctop 19966   clsccl 20082   neicnei 20162    fLimf cflf 20999  CnExtccnext 21123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  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-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-pm 7501  df-cnext 21124
This theorem is referenced by:  cnextfun  21128
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