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Theorem iscn 19903
Description: The predicate " F is a continuous function from topology  J to topology  K." Definition of continuous function in [Munkres] p. 102. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
iscn  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( F  e.  ( J  Cn  K
)  <->  ( 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 iscn
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 cnfval 19901 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( J  Cn  K )  =  {
f  e.  ( Y  ^m  X )  | 
A. y  e.  K  ( `' f " y
)  e.  J }
)
21eleq2d 2524 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( F  e.  ( J  Cn  K
)  <->  F  e.  { f  e.  ( Y  ^m  X )  |  A. y  e.  K  ( `' f " y
)  e.  J }
) )
3 cnveq 5165 . . . . . . 7  |-  ( f  =  F  ->  `' f  =  `' F
)
43imaeq1d 5324 . . . . . 6  |-  ( f  =  F  ->  ( `' f " y
)  =  ( `' F " y ) )
54eleq1d 2523 . . . . 5  |-  ( f  =  F  ->  (
( `' f "
y )  e.  J  <->  ( `' F " y )  e.  J ) )
65ralbidv 2893 . . . 4  |-  ( f  =  F  ->  ( A. y  e.  K  ( `' f " y
)  e.  J  <->  A. y  e.  K  ( `' F " y )  e.  J ) )
76elrab 3254 . . 3  |-  ( F  e.  { f  e.  ( Y  ^m  X
)  |  A. y  e.  K  ( `' f " y )  e.  J }  <->  ( F  e.  ( Y  ^m  X
)  /\  A. y  e.  K  ( `' F " y )  e.  J ) )
8 toponmax 19596 . . . . 5  |-  ( K  e.  (TopOn `  Y
)  ->  Y  e.  K )
9 toponmax 19596 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
10 elmapg 7425 . . . . 5  |-  ( ( Y  e.  K  /\  X  e.  J )  ->  ( F  e.  ( Y  ^m  X )  <-> 
F : X --> Y ) )
118, 9, 10syl2anr 476 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( F  e.  ( Y  ^m  X
)  <->  F : X --> Y ) )
1211anbi1d 702 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( ( F  e.  ( Y  ^m  X )  /\  A. y  e.  K  ( `' F " y )  e.  J )  <->  ( F : X --> Y  /\  A. y  e.  K  ( `' F " y )  e.  J ) ) )
137, 12syl5bb 257 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( F  e.  { f  e.  ( Y  ^m  X )  |  A. y  e.  K  ( `' f
" y )  e.  J }  <->  ( F : X --> Y  /\  A. y  e.  K  ( `' F " y )  e.  J ) ) )
142, 13bitrd 253 1  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( F  e.  ( J  Cn  K
)  <->  ( F : X
--> Y  /\  A. y  e.  K  ( `' F " y )  e.  J ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   {crab 2808   `'ccnv 4987   "cima 4991   -->wf 5566   ` cfv 5570  (class class class)co 6270    ^m cmap 7412  TopOnctopon 19562    Cn ccn 19892
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-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-rab 2813  df-v 3108  df-sbc 3325  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-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-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-map 7414  df-top 19566  df-topon 19569  df-cn 19895
This theorem is referenced by:  iscn2  19906  cnf2  19917  tgcn  19920  ssidcn  19923  iscncl  19937  cnntr  19943  cnss1  19944  cnss2  19945  cncnp  19948  cnrest  19953  cnrest2  19954  cndis  19959  cnindis  19960  kgencn  20223  kgencn3  20225  tx1cn  20276  tx2cn  20277  txdis1cn  20302  qtopid  20372  qtopcn  20381  qtopf1  20483  qustgplem  20785  ucncn  20954  cvmlift2lem9a  29012  rfcnpre1  31634  0cnf  31918
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