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Theorem cnextrel 20732
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 5098 . . . 4  |-  Rel  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )
21rgenw 2815 . . 3  |-  A. x  e.  ( ( cls `  J
) `  A ) Rel  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )
3 reliun 5111 . . 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 209 . 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 20731 . . 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 5075 . 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 233 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 367    = wceq 1398    e. wcel 1823   A.wral 2804    C_ wss 3461   {csn 4016   U.cuni 4235   U_ciun 4315    X. cxp 4986   Rel wrel 4993   -->wf 5566   ` cfv 5570  (class class class)co 6270   ↾t crest 14913   Topctop 19564   clsccl 19689   neicnei 19768    fLimf cflf 20605  CnExtccnext 20728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-pm 7415  df-cnext 20729
This theorem is referenced by:  cnextfun  20733
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