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Theorem icnpimaex 12307
Description: Property of a function continuous at a point. (Contributed by FL, 31-Dec-2006.) (Revised by Jim Kingdon, 28-Mar-2023.)
Assertion
Ref Expression
icnpimaex  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  P  e.  X
)  /\  ( F  e.  ( ( J  CnP  K ) `  P )  /\  A  e.  K  /\  ( F `  P
)  e.  A ) )  ->  E. x  e.  J  ( P  e.  x  /\  ( F " x )  C_  A ) )
Distinct variable groups:    x, A    x, F    x, J    x, K    x, P    x, X    x, Y

Proof of Theorem icnpimaex
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 simpr3 974 . 2  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  P  e.  X
)  /\  ( F  e.  ( ( J  CnP  K ) `  P )  /\  A  e.  K  /\  ( F `  P
)  e.  A ) )  ->  ( F `  P )  e.  A
)
2 eleq2 2181 . . . 4  |-  ( y  =  A  ->  (
( F `  P
)  e.  y  <->  ( F `  P )  e.  A
) )
3 sseq2 3091 . . . . . 6  |-  ( y  =  A  ->  (
( F " x
)  C_  y  <->  ( F " x )  C_  A
) )
43anbi2d 459 . . . . 5  |-  ( y  =  A  ->  (
( P  e.  x  /\  ( F " x
)  C_  y )  <->  ( P  e.  x  /\  ( F " x ) 
C_  A ) ) )
54rexbidv 2415 . . . 4  |-  ( y  =  A  ->  ( E. x  e.  J  ( P  e.  x  /\  ( F " x
)  C_  y )  <->  E. x  e.  J  ( P  e.  x  /\  ( F " x ) 
C_  A ) ) )
62, 5imbi12d 233 . . 3  |-  ( y  =  A  ->  (
( ( F `  P )  e.  y  ->  E. x  e.  J  ( P  e.  x  /\  ( F " x
)  C_  y )
)  <->  ( ( F `
 P )  e.  A  ->  E. x  e.  J  ( P  e.  x  /\  ( F " x )  C_  A ) ) ) )
7 simpr1 972 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  P  e.  X
)  /\  ( F  e.  ( ( J  CnP  K ) `  P )  /\  A  e.  K  /\  ( F `  P
)  e.  A ) )  ->  F  e.  ( ( J  CnP  K ) `  P ) )
8 iscnp 12295 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  P  e.  X
)  ->  ( F  e.  ( ( J  CnP  K ) `  P )  <-> 
( F : X --> Y  /\  A. y  e.  K  ( ( F `
 P )  e.  y  ->  E. x  e.  J  ( P  e.  x  /\  ( F " x )  C_  y ) ) ) ) )
98adantr 274 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  P  e.  X
)  /\  ( F  e.  ( ( J  CnP  K ) `  P )  /\  A  e.  K  /\  ( F `  P
)  e.  A ) )  ->  ( F  e.  ( ( J  CnP  K ) `  P )  <-> 
( F : X --> Y  /\  A. y  e.  K  ( ( F `
 P )  e.  y  ->  E. x  e.  J  ( P  e.  x  /\  ( F " x )  C_  y ) ) ) ) )
107, 9mpbid 146 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  P  e.  X
)  /\  ( F  e.  ( ( J  CnP  K ) `  P )  /\  A  e.  K  /\  ( F `  P
)  e.  A ) )  ->  ( F : X --> Y  /\  A. y  e.  K  (
( F `  P
)  e.  y  ->  E. x  e.  J  ( P  e.  x  /\  ( F " x
)  C_  y )
) ) )
1110simprd 113 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  P  e.  X
)  /\  ( F  e.  ( ( J  CnP  K ) `  P )  /\  A  e.  K  /\  ( F `  P
)  e.  A ) )  ->  A. y  e.  K  ( ( F `  P )  e.  y  ->  E. x  e.  J  ( P  e.  x  /\  ( F " x )  C_  y ) ) )
12 simpr2 973 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  P  e.  X
)  /\  ( F  e.  ( ( J  CnP  K ) `  P )  /\  A  e.  K  /\  ( F `  P
)  e.  A ) )  ->  A  e.  K )
136, 11, 12rspcdva 2768 . 2  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  P  e.  X
)  /\  ( F  e.  ( ( J  CnP  K ) `  P )  /\  A  e.  K  /\  ( F `  P
)  e.  A ) )  ->  ( ( F `  P )  e.  A  ->  E. x  e.  J  ( P  e.  x  /\  ( F " x )  C_  A ) ) )
141, 13mpd 13 1  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  P  e.  X
)  /\  ( F  e.  ( ( J  CnP  K ) `  P )  /\  A  e.  K  /\  ( F `  P
)  e.  A ) )  ->  E. x  e.  J  ( P  e.  x  /\  ( F " x )  C_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 947    = wceq 1316    e. wcel 1465   A.wral 2393   E.wrex 2394    C_ wss 3041   "cima 4512   -->wf 5089   ` cfv 5093  (class class class)co 5742  TopOnctopon 12104    CnP ccnp 12282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-map 6512  df-top 12092  df-topon 12105  df-cnp 12285
This theorem is referenced by:  iscnp4  12314  cnpnei  12315  cnptopco  12318  cncnp  12326  cnptopresti  12334  lmtopcnp  12346  txcnp  12367
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