ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  resflem Unicode version

Theorem resflem 5550
Description: A lemma to bound the range of a restriction. The conclusion would also hold with  ( X  i^i  Y ) in place of  Y (provided  x does not occur in  X). If that stronger result is needed, it is however simpler to use the instance of resflem 5550 where  ( X  i^i  Y ) is substituted for  Y (in both the conclusion and the third hypothesis). (Contributed by BJ, 4-Jul-2022.)
Hypotheses
Ref Expression
resflem.1  |-  ( ph  ->  F : V --> X )
resflem.2  |-  ( ph  ->  A  C_  V )
resflem.3  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  e.  Y )
Assertion
Ref Expression
resflem  |-  ( ph  ->  ( F  |`  A ) : A --> Y )
Distinct variable groups:    x, A    ph, x    x, F    x, Y
Allowed substitution hints:    V( x)    X( x)

Proof of Theorem resflem
StepHypRef Expression
1 resflem.2 . . . . . 6  |-  ( ph  ->  A  C_  V )
21sseld 3064 . . . . 5  |-  ( ph  ->  ( x  e.  A  ->  x  e.  V ) )
3 resflem.1 . . . . . . 7  |-  ( ph  ->  F : V --> X )
4 fdm 5246 . . . . . . 7  |-  ( F : V --> X  ->  dom  F  =  V )
53, 4syl 14 . . . . . 6  |-  ( ph  ->  dom  F  =  V )
65eleq2d 2185 . . . . 5  |-  ( ph  ->  ( x  e.  dom  F  <-> 
x  e.  V ) )
72, 6sylibrd 168 . . . 4  |-  ( ph  ->  ( x  e.  A  ->  x  e.  dom  F
) )
8 resflem.3 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  e.  Y )
98ex 114 . . . 4  |-  ( ph  ->  ( x  e.  A  ->  ( F `  x
)  e.  Y ) )
107, 9jcad 303 . . 3  |-  ( ph  ->  ( x  e.  A  ->  ( x  e.  dom  F  /\  ( F `  x )  e.  Y
) ) )
1110ralrimiv 2479 . 2  |-  ( ph  ->  A. x  e.  A  ( x  e.  dom  F  /\  ( F `  x )  e.  Y
) )
12 ffun 5243 . . . 4  |-  ( F : V --> X  ->  Fun  F )
133, 12syl 14 . . 3  |-  ( ph  ->  Fun  F )
14 ffvresb 5549 . . 3  |-  ( Fun 
F  ->  ( ( F  |`  A ) : A --> Y  <->  A. x  e.  A  ( x  e.  dom  F  /\  ( F `  x )  e.  Y ) ) )
1513, 14syl 14 . 2  |-  ( ph  ->  ( ( F  |`  A ) : A --> Y 
<-> 
A. x  e.  A  ( x  e.  dom  F  /\  ( F `  x )  e.  Y
) ) )
1611, 15mpbird 166 1  |-  ( ph  ->  ( F  |`  A ) : A --> Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463   A.wral 2391    C_ wss 3039   dom cdm 4507    |` cres 4509   Fun wfun 5085   -->wf 5087   ` cfv 5091
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-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-fv 5099
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator