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Theorem resflem 5631
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 5631 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 3127 . . . . 5  |-  ( ph  ->  ( x  e.  A  ->  x  e.  V ) )
3 resflem.1 . . . . . . 7  |-  ( ph  ->  F : V --> X )
4 fdm 5325 . . . . . . 7  |-  ( F : V --> X  ->  dom  F  =  V )
53, 4syl 14 . . . . . 6  |-  ( ph  ->  dom  F  =  V )
65eleq2d 2227 . . . . 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 305 . . 3  |-  ( ph  ->  ( x  e.  A  ->  ( x  e.  dom  F  /\  ( F `  x )  e.  Y
) ) )
1110ralrimiv 2529 . 2  |-  ( ph  ->  A. x  e.  A  ( x  e.  dom  F  /\  ( F `  x )  e.  Y
) )
12 ffun 5322 . . . 4  |-  ( F : V --> X  ->  Fun  F )
133, 12syl 14 . . 3  |-  ( ph  ->  Fun  F )
14 ffvresb 5630 . . 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 1335    e. wcel 2128   A.wral 2435    C_ wss 3102   dom cdm 4586    |` cres 4588   Fun wfun 5164   -->wf 5166   ` cfv 5170
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-fv 5178
This theorem is referenced by:  bj-charfun  13393
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