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Theorem dfimafnf 23996
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 3302 . . . . . . 7  |-  ( A 
C_  dom  F  ->  ( z  e.  A  -> 
z  e.  dom  F
) )
2 eqcom 2406 . . . . . . . . 9  |-  ( ( F `  z )  =  y  <->  y  =  ( F `  z ) )
3 funbrfvb 5728 . . . . . . . . 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 2683 . . . 4  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( E. z  e.  A  y  =  ( F `  z )  <->  E. z  e.  A  z F y ) )
98abbidv 2518 . . 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 5164 . . 3  |-  ( F
" A )  =  { y  |  E. z  e.  A  z F y }
119, 10syl6reqr 2455 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F " A
)  =  { y  |  E. z  e.  A  y  =  ( F `  z ) } )
12 nfcv 2540 . . . 4  |-  F/_ z A
13 dfimafnf.1 . . . 4  |-  F/_ x A
14 dfimafnf.2 . . . . . 6  |-  F/_ x F
15 nfcv 2540 . . . . . 6  |-  F/_ x
z
1614, 15nffv 5694 . . . . 5  |-  F/_ x
( F `  z
)
1716nfeq2 2551 . . . 4  |-  F/ x  y  =  ( F `  z )
18 nfv 1626 . . . 4  |-  F/ z  y  =  ( F `
 x )
19 fveq2 5687 . . . . 5  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
2019eqeq2d 2415 . . . 4  |-  ( z  =  x  ->  (
y  =  ( F `
 z )  <->  y  =  ( F `  x ) ) )
2112, 13, 17, 18, 20cbvrexf 2887 . . 3  |-  ( E. z  e.  A  y  =  ( F `  z )  <->  E. x  e.  A  y  =  ( F `  x ) )
2221abbii 2516 . 2  |-  { y  |  E. z  e.  A  y  =  ( F `  z ) }  =  { y  |  E. x  e.  A  y  =  ( F `  x ) }
2311, 22syl6eq 2452 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 1649    e. wcel 1721   {cab 2390   F/_wnfc 2527   E.wrex 2667    C_ wss 3280   class class class wbr 4172   dom cdm 4837   "cima 4840   Fun wfun 5407   ` cfv 5413
This theorem is referenced by:  funimass4f  23997
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-fv 5421
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