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Theorem dfimafnf 23057
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 3187 . . . . . . 7  |-  ( A 
C_  dom  F  ->  ( z  e.  A  -> 
z  e.  dom  F
) )
2 eqcom 2298 . . . . . . . . 9  |-  ( ( F `  z )  =  y  <->  y  =  ( F `  z ) )
3 funbrfvb 5581 . . . . . . . . 9  |-  ( ( Fun  F  /\  z  e.  dom  F )  -> 
( ( F `  z )  =  y  <-> 
z F y ) )
42, 3syl5bbr 250 . . . . . . . 8  |-  ( ( Fun  F  /\  z  e.  dom  F )  -> 
( y  =  ( F `  z )  <-> 
z F y ) )
54ex 423 . . . . . . 7  |-  ( Fun 
F  ->  ( z  e.  dom  F  ->  (
y  =  ( F `
 z )  <->  z F
y ) ) )
61, 5syl9r 67 . . . . . 6  |-  ( Fun 
F  ->  ( A  C_ 
dom  F  ->  ( z  e.  A  ->  (
y  =  ( F `
 z )  <->  z F
y ) ) ) )
76imp31 421 . . . . 5  |-  ( ( ( Fun  F  /\  A  C_  dom  F )  /\  z  e.  A
)  ->  ( y  =  ( F `  z )  <->  z F
y ) )
87rexbidva 2573 . . . 4  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( E. z  e.  A  y  =  ( F `  z )  <->  E. z  e.  A  z F y ) )
98abbidv 2410 . . 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 5030 . . 3  |-  ( F
" A )  =  { y  |  E. z  e.  A  z F y }
119, 10syl6reqr 2347 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F " A
)  =  { y  |  E. z  e.  A  y  =  ( F `  z ) } )
12 nfcv 2432 . . . 4  |-  F/_ z A
13 dfimafnf.1 . . . 4  |-  F/_ x A
14 nfcv 2432 . . . . 5  |-  F/_ x
y
15 dfimafnf.2 . . . . . 6  |-  F/_ x F
16 nfcv 2432 . . . . . 6  |-  F/_ x
z
1715, 16nffv 5548 . . . . 5  |-  F/_ x
( F `  z
)
1814, 17nfeq 2439 . . . 4  |-  F/ x  y  =  ( F `  z )
19 nfv 1609 . . . 4  |-  F/ z  y  =  ( F `
 x )
20 fveq2 5541 . . . . 5  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
2120eqeq2d 2307 . . . 4  |-  ( z  =  x  ->  (
y  =  ( F `
 z )  <->  y  =  ( F `  x ) ) )
2212, 13, 18, 19, 21cbvrexf 2772 . . 3  |-  ( E. z  e.  A  y  =  ( F `  z )  <->  E. x  e.  A  y  =  ( F `  x ) )
2322abbii 2408 . 2  |-  { y  |  E. z  e.  A  y  =  ( F `  z ) }  =  { y  |  E. x  e.  A  y  =  ( F `  x ) }
2411, 23syl6eq 2344 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   F/_wnfc 2419   E.wrex 2557    C_ wss 3165   class class class wbr 4039   dom cdm 4705   "cima 4708   Fun wfun 5265   ` cfv 5271
This theorem is referenced by:  funimass4f  23058
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279
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