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Theorem dfimafnf 24035
Description: Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.) (Revised by Thierry Arnoux, 24-Apr-2017.)
Hypotheses
Ref Expression
dfimafnf.1  |-  F/_ x A
dfimafnf.2  |-  F/_ x F
Assertion
Ref Expression
dfimafnf  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F " A
)  =  { y  |  E. x  e.  A  y  =  ( F `  x ) } )
Distinct variable groups:    x, y    y, A    y, F
Allowed substitution hints:    A( x)    F( x)

Proof of Theorem dfimafnf
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 ssel 3334 . . . . . . 7  |-  ( A 
C_  dom  F  ->  ( z  e.  A  -> 
z  e.  dom  F
) )
2 eqcom 2437 . . . . . . . . 9  |-  ( ( F `  z )  =  y  <->  y  =  ( F `  z ) )
3 funbrfvb 5761 . . . . . . . . 9  |-  ( ( Fun  F  /\  z  e.  dom  F )  -> 
( ( F `  z )  =  y  <-> 
z F y ) )
42, 3syl5bbr 251 . . . . . . . 8  |-  ( ( Fun  F  /\  z  e.  dom  F )  -> 
( y  =  ( F `  z )  <-> 
z F y ) )
54ex 424 . . . . . . 7  |-  ( Fun 
F  ->  ( z  e.  dom  F  ->  (
y  =  ( F `
 z )  <->  z F
y ) ) )
61, 5syl9r 69 . . . . . 6  |-  ( Fun 
F  ->  ( A  C_ 
dom  F  ->  ( z  e.  A  ->  (
y  =  ( F `
 z )  <->  z F
y ) ) ) )
76imp31 422 . . . . 5  |-  ( ( ( Fun  F  /\  A  C_  dom  F )  /\  z  e.  A
)  ->  ( y  =  ( F `  z )  <->  z F
y ) )
87rexbidva 2714 . . . 4  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( E. z  e.  A  y  =  ( F `  z )  <->  E. z  e.  A  z F y ) )
98abbidv 2549 . . 3  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  ->  { y  |  E. z  e.  A  y  =  ( F `  z ) }  =  { y  |  E. z  e.  A  z F y } )
10 dfima2 5197 . . 3  |-  ( F
" A )  =  { y  |  E. z  e.  A  z F y }
119, 10syl6reqr 2486 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F " A
)  =  { y  |  E. z  e.  A  y  =  ( F `  z ) } )
12 nfcv 2571 . . . 4  |-  F/_ z A
13 dfimafnf.1 . . . 4  |-  F/_ x A
14 dfimafnf.2 . . . . . 6  |-  F/_ x F
15 nfcv 2571 . . . . . 6  |-  F/_ x
z
1614, 15nffv 5727 . . . . 5  |-  F/_ x
( F `  z
)
1716nfeq2 2582 . . . 4  |-  F/ x  y  =  ( F `  z )
18 nfv 1629 . . . 4  |-  F/ z  y  =  ( F `
 x )
19 fveq2 5720 . . . . 5  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
2019eqeq2d 2446 . . . 4  |-  ( z  =  x  ->  (
y  =  ( F `
 z )  <->  y  =  ( F `  x ) ) )
2112, 13, 17, 18, 20cbvrexf 2919 . . 3  |-  ( E. z  e.  A  y  =  ( F `  z )  <->  E. x  e.  A  y  =  ( F `  x ) )
2221abbii 2547 . 2  |-  { y  |  E. z  e.  A  y  =  ( F `  z ) }  =  { y  |  E. x  e.  A  y  =  ( F `  x ) }
2311, 22syl6eq 2483 1  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F " A
)  =  { y  |  E. x  e.  A  y  =  ( F `  x ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2421   F/_wnfc 2558   E.wrex 2698    C_ wss 3312   class class class wbr 4204   dom cdm 4870   "cima 4873   Fun wfun 5440   ` cfv 5446
This theorem is referenced by:  funimass4f  24036
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454
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