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Theorem resflem 5462
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 5462 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 3024 . . . . 5  |-  ( ph  ->  ( x  e.  A  ->  x  e.  V ) )
3 resflem.1 . . . . . . 7  |-  ( ph  ->  F : V --> X )
4 fdm 5166 . . . . . . 7  |-  ( F : V --> X  ->  dom  F  =  V )
53, 4syl 14 . . . . . 6  |-  ( ph  ->  dom  F  =  V )
65eleq2d 2157 . . . . 5  |-  ( ph  ->  ( x  e.  dom  F  <-> 
x  e.  V ) )
72, 6sylibrd 167 . . . 4  |-  ( ph  ->  ( x  e.  A  ->  x  e.  dom  F
) )
8 resflem.3 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  e.  Y )
98ex 113 . . . 4  |-  ( ph  ->  ( x  e.  A  ->  ( F `  x
)  e.  Y ) )
107, 9jcad 301 . . 3  |-  ( ph  ->  ( x  e.  A  ->  ( x  e.  dom  F  /\  ( F `  x )  e.  Y
) ) )
1110ralrimiv 2445 . 2  |-  ( ph  ->  A. x  e.  A  ( x  e.  dom  F  /\  ( F `  x )  e.  Y
) )
12 ffun 5164 . . . 4  |-  ( F : V --> X  ->  Fun  F )
133, 12syl 14 . . 3  |-  ( ph  ->  Fun  F )
14 ffvresb 5461 . . 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 165 1  |-  ( ph  ->  ( F  |`  A ) : A --> Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   A.wral 2359    C_ wss 2999   dom cdm 4438    |` cres 4440   Fun wfun 5009   -->wf 5011   ` cfv 5015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fv 5023
This theorem is referenced by: (None)
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