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Theorem cnextrel 19737
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 5031 . . . 4  |-  Rel  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )
21rgenw 2869 . . 3  |-  A. x  e.  ( ( cls `  J
) `  A ) Rel  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )
3 reliun 5044 . . 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 19736 . . 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 5008 . 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 369    = wceq 1370    e. wcel 1757   A.wral 2792    C_ wss 3412   {csn 3961   U.cuni 4175   U_ciun 4255    X. cxp 4922   Rel wrel 4929   -->wf 5498   ` cfv 5502  (class class class)co 6176   ↾t crest 14447   Topctop 18600   clsccl 18724   neicnei 18803    fLimf cflf 19610  CnExtccnext 19733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-pm 7303  df-cnext 19734
This theorem is referenced by:  cnextfun  19738
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