MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iscn2 Structured version   Unicode version

Theorem iscn2 19500
Description: The predicate " F is a continuous function from topology  J to topology  K." Definition of continuous function in [Munkres] p. 102. (Contributed by Mario Carneiro, 21-Aug-2015.)
Hypotheses
Ref Expression
iscn.1  |-  X  = 
U. J
iscn.2  |-  Y  = 
U. K
Assertion
Ref Expression
iscn2  |-  ( F  e.  ( J  Cn  K )  <->  ( ( J  e.  Top  /\  K  e.  Top )  /\  ( F : X --> Y  /\  A. y  e.  K  ( `' F " y )  e.  J ) ) )
Distinct variable groups:    y, J    y, K    y, X    y, F    y, Y

Proof of Theorem iscn2
Dummy variables  f 
j  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cn 19489 . . 3  |-  Cn  =  ( j  e.  Top ,  k  e.  Top  |->  { f  e.  ( U. k  ^m  U. j )  |  A. y  e.  k  ( `' f
" y )  e.  j } )
21elmpt2cl 6494 . 2  |-  ( F  e.  ( J  Cn  K )  ->  ( J  e.  Top  /\  K  e.  Top ) )
3 iscn.1 . . . 4  |-  X  = 
U. J
43toptopon 19196 . . 3  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
5 iscn.2 . . . 4  |-  Y  = 
U. K
65toptopon 19196 . . 3  |-  ( K  e.  Top  <->  K  e.  (TopOn `  Y ) )
7 iscn 19497 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( F  e.  ( J  Cn  K
)  <->  ( F : X
--> Y  /\  A. y  e.  K  ( `' F " y )  e.  J ) ) )
84, 6, 7syl2anb 479 . 2  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( F  e.  ( J  Cn  K )  <-> 
( F : X --> Y  /\  A. y  e.  K  ( `' F " y )  e.  J
) ) )
92, 8biadan2 642 1  |-  ( F  e.  ( J  Cn  K )  <->  ( ( J  e.  Top  /\  K  e.  Top )  /\  ( F : X --> Y  /\  A. y  e.  K  ( `' F " y )  e.  J ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2809   {crab 2813   U.cuni 4240   `'ccnv 4993   "cima 4997   -->wf 5577   ` cfv 5581  (class class class)co 6277    ^m cmap 7412   Topctop 19156  TopOnctopon 19157    Cn ccn 19486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-map 7414  df-top 19161  df-topon 19164  df-cn 19489
This theorem is referenced by:  cntop1  19502  cntop2  19503  cnf  19508  cnima  19527  cnco  19528  ptpjcn  19842
  Copyright terms: Public domain W3C validator