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Theorem dfimafnf 23887
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 3286 . . . . . . 7  |-  ( A 
C_  dom  F  ->  ( z  e.  A  -> 
z  e.  dom  F
) )
2 eqcom 2390 . . . . . . . . 9  |-  ( ( F `  z )  =  y  <->  y  =  ( F `  z ) )
3 funbrfvb 5709 . . . . . . . . 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 2667 . . . 4  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( E. z  e.  A  y  =  ( F `  z )  <->  E. z  e.  A  z F y ) )
98abbidv 2502 . . 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 5146 . . 3  |-  ( F
" A )  =  { y  |  E. z  e.  A  z F y }
119, 10syl6reqr 2439 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F " A
)  =  { y  |  E. z  e.  A  y  =  ( F `  z ) } )
12 nfcv 2524 . . . 4  |-  F/_ z A
13 dfimafnf.1 . . . 4  |-  F/_ x A
14 dfimafnf.2 . . . . . 6  |-  F/_ x F
15 nfcv 2524 . . . . . 6  |-  F/_ x
z
1614, 15nffv 5676 . . . . 5  |-  F/_ x
( F `  z
)
1716nfeq2 2535 . . . 4  |-  F/ x  y  =  ( F `  z )
18 nfv 1626 . . . 4  |-  F/ z  y  =  ( F `
 x )
19 fveq2 5669 . . . . 5  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
2019eqeq2d 2399 . . . 4  |-  ( z  =  x  ->  (
y  =  ( F `
 z )  <->  y  =  ( F `  x ) ) )
2112, 13, 17, 18, 20cbvrexf 2871 . . 3  |-  ( E. z  e.  A  y  =  ( F `  z )  <->  E. x  e.  A  y  =  ( F `  x ) )
2221abbii 2500 . 2  |-  { y  |  E. z  e.  A  y  =  ( F `  z ) }  =  { y  |  E. x  e.  A  y  =  ( F `  x ) }
2311, 22syl6eq 2436 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 1717   {cab 2374   F/_wnfc 2511   E.wrex 2651    C_ wss 3264   class class class wbr 4154   dom cdm 4819   "cima 4822   Fun wfun 5389   ` cfv 5395
This theorem is referenced by:  funimass4f  23888
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-fv 5403
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