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Theorem icnpimaex 13344
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 1005 . 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 2241 . . . 4  |-  ( y  =  A  ->  (
( F `  P
)  e.  y  <->  ( F `  P )  e.  A
) )
3 sseq2 3179 . . . . . 6  |-  ( y  =  A  ->  (
( F " x
)  C_  y  <->  ( F " x )  C_  A
) )
43anbi2d 464 . . . . 5  |-  ( y  =  A  ->  (
( P  e.  x  /\  ( F " x
)  C_  y )  <->  ( P  e.  x  /\  ( F " x ) 
C_  A ) ) )
54rexbidv 2478 . . . 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 234 . . 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 1003 . . . . 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 13332 . . . . . 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 276 . . . . 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 147 . . . 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 114 . . 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 1004 . . 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 2846 . 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 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   A.wral 2455   E.wrex 2456    C_ wss 3129   "cima 4625   -->wf 5207   ` cfv 5211  (class class class)co 5868  TopOnctopon 13141    CnP ccnp 13319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-map 6643  df-top 13129  df-topon 13142  df-cnp 13322
This theorem is referenced by:  iscnp4  13351  cnpnei  13352  cnptopco  13355  cncnp  13363  cnptopresti  13371  lmtopcnp  13383  txcnp  13404
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