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Theorem dfimafnf 23041
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 3174 . . . . . . 7  |-  ( A 
C_  dom  F  ->  ( z  e.  A  -> 
z  e.  dom  F
) )
2 eqcom 2285 . . . . . . . . 9  |-  ( ( F `  z )  =  y  <->  y  =  ( F `  z ) )
3 funbrfvb 5565 . . . . . . . . 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 2560 . . . 4  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( E. z  e.  A  y  =  ( F `  z )  <->  E. z  e.  A  z F y ) )
98abbidv 2397 . . 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 5014 . . 3  |-  ( F
" A )  =  { y  |  E. z  e.  A  z F y }
119, 10syl6reqr 2334 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F " A
)  =  { y  |  E. z  e.  A  y  =  ( F `  z ) } )
12 nfcv 2419 . . . 4  |-  F/_ z A
13 dfimafnf.1 . . . 4  |-  F/_ x A
14 nfcv 2419 . . . . 5  |-  F/_ x
y
15 dfimafnf.2 . . . . . 6  |-  F/_ x F
16 nfcv 2419 . . . . . 6  |-  F/_ x
z
1715, 16nffv 5532 . . . . 5  |-  F/_ x
( F `  z
)
1814, 17nfeq 2426 . . . 4  |-  F/ x  y  =  ( F `  z )
19 nfv 1605 . . . 4  |-  F/ z  y  =  ( F `
 x )
20 fveq2 5525 . . . . 5  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
2120eqeq2d 2294 . . . 4  |-  ( z  =  x  ->  (
y  =  ( F `
 z )  <->  y  =  ( F `  x ) ) )
2212, 13, 18, 19, 21cbvrexf 2759 . . 3  |-  ( E. z  e.  A  y  =  ( F `  z )  <->  E. x  e.  A  y  =  ( F `  x ) )
2322abbii 2395 . 2  |-  { y  |  E. z  e.  A  y  =  ( F `  z ) }  =  { y  |  E. x  e.  A  y  =  ( F `  x ) }
2411, 23syl6eq 2331 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 1623    e. wcel 1684   {cab 2269   F/_wnfc 2406   E.wrex 2544    C_ wss 3152   class class class wbr 4023   dom cdm 4689   "cima 4692   Fun wfun 5249   ` cfv 5255
This theorem is referenced by:  funimass4f  23042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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