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Theorem fvelimab 5257
Description: Function value in an image. (Contributed by NM, 20-Jan-2007.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by David Abernethy, 17-Dec-2011.)
Assertion
Ref Expression
fvelimab  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( C  e.  ( F " B )  <->  E. x  e.  B  ( F `  x )  =  C ) )
Distinct variable groups:    x, B    x, C    x, F
Allowed substitution hint:    A( x)

Proof of Theorem fvelimab
Dummy variables  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2583 . . . 4  |-  ( C  e.  ( F " B )  ->  C  e.  _V )
21anim2i 328 . . 3  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  C  e.  ( F " B ) )  ->  ( ( F  Fn  A  /\  B  C_  A )  /\  C  e.  _V )
)
3 ssel2 2968 . . . . . . . 8  |-  ( ( B  C_  A  /\  u  e.  B )  ->  u  e.  A )
4 funfvex 5220 . . . . . . . . 9  |-  ( ( Fun  F  /\  u  e.  dom  F )  -> 
( F `  u
)  e.  _V )
54funfni 5027 . . . . . . . 8  |-  ( ( F  Fn  A  /\  u  e.  A )  ->  ( F `  u
)  e.  _V )
63, 5sylan2 274 . . . . . . 7  |-  ( ( F  Fn  A  /\  ( B  C_  A  /\  u  e.  B )
)  ->  ( F `  u )  e.  _V )
76anassrs 386 . . . . . 6  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  u  e.  B
)  ->  ( F `  u )  e.  _V )
8 eleq1 2116 . . . . . 6  |-  ( ( F `  u )  =  C  ->  (
( F `  u
)  e.  _V  <->  C  e.  _V ) )
97, 8syl5ibcom 148 . . . . 5  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  u  e.  B
)  ->  ( ( F `  u )  =  C  ->  C  e. 
_V ) )
109rexlimdva 2450 . . . 4  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( E. u  e.  B  ( F `  u )  =  C  ->  C  e.  _V ) )
1110imdistani 427 . . 3  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  E. u  e.  B  ( F `  u )  =  C )  ->  ( ( F  Fn  A  /\  B  C_  A )  /\  C  e.  _V )
)
12 eleq1 2116 . . . . . . 7  |-  ( v  =  C  ->  (
v  e.  ( F
" B )  <->  C  e.  ( F " B ) ) )
13 eqeq2 2065 . . . . . . . 8  |-  ( v  =  C  ->  (
( F `  u
)  =  v  <->  ( F `  u )  =  C ) )
1413rexbidv 2344 . . . . . . 7  |-  ( v  =  C  ->  ( E. u  e.  B  ( F `  u )  =  v  <->  E. u  e.  B  ( F `  u )  =  C ) )
1512, 14bibi12d 228 . . . . . 6  |-  ( v  =  C  ->  (
( v  e.  ( F " B )  <->  E. u  e.  B  ( F `  u )  =  v )  <->  ( C  e.  ( F " B
)  <->  E. u  e.  B  ( F `  u )  =  C ) ) )
1615imbi2d 223 . . . . 5  |-  ( v  =  C  ->  (
( ( F  Fn  A  /\  B  C_  A
)  ->  ( v  e.  ( F " B
)  <->  E. u  e.  B  ( F `  u )  =  v ) )  <-> 
( ( F  Fn  A  /\  B  C_  A
)  ->  ( C  e.  ( F " B
)  <->  E. u  e.  B  ( F `  u )  =  C ) ) ) )
17 fnfun 5024 . . . . . . . 8  |-  ( F  Fn  A  ->  Fun  F )
1817adantr 265 . . . . . . 7  |-  ( ( F  Fn  A  /\  B  C_  A )  ->  Fun  F )
19 fndm 5026 . . . . . . . . 9  |-  ( F  Fn  A  ->  dom  F  =  A )
2019sseq2d 3001 . . . . . . . 8  |-  ( F  Fn  A  ->  ( B  C_  dom  F  <->  B  C_  A
) )
2120biimpar 285 . . . . . . 7  |-  ( ( F  Fn  A  /\  B  C_  A )  ->  B  C_  dom  F )
22 dfimafn 5250 . . . . . . 7  |-  ( ( Fun  F  /\  B  C_ 
dom  F )  -> 
( F " B
)  =  { v  |  E. u  e.  B  ( F `  u )  =  v } )
2318, 21, 22syl2anc 397 . . . . . 6  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( F " B
)  =  { v  |  E. u  e.  B  ( F `  u )  =  v } )
2423abeq2d 2166 . . . . 5  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( v  e.  ( F " B )  <->  E. u  e.  B  ( F `  u )  =  v ) )
2516, 24vtoclg 2630 . . . 4  |-  ( C  e.  _V  ->  (
( F  Fn  A  /\  B  C_  A )  ->  ( C  e.  ( F " B
)  <->  E. u  e.  B  ( F `  u )  =  C ) ) )
2625impcom 120 . . 3  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  C  e.  _V )  ->  ( C  e.  ( F " B
)  <->  E. u  e.  B  ( F `  u )  =  C ) )
272, 11, 26pm5.21nd 836 . 2  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( C  e.  ( F " B )  <->  E. u  e.  B  ( F `  u )  =  C ) )
28 fveq2 5206 . . . 4  |-  ( u  =  x  ->  ( F `  u )  =  ( F `  x ) )
2928eqeq1d 2064 . . 3  |-  ( u  =  x  ->  (
( F `  u
)  =  C  <->  ( F `  x )  =  C ) )
3029cbvrexv 2551 . 2  |-  ( E. u  e.  B  ( F `  u )  =  C  <->  E. x  e.  B  ( F `  x )  =  C )
3127, 30syl6bb 189 1  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( C  e.  ( F " B )  <->  E. x  e.  B  ( F `  x )  =  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1259    e. wcel 1409   {cab 2042   E.wrex 2324   _Vcvv 2574    C_ wss 2945   dom cdm 4373   "cima 4376   Fun wfun 4924    Fn wfn 4925   ` cfv 4930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-fv 4938
This theorem is referenced by:  ssimaex  5262  rexima  5422  ralima  5423  f1elima  5440  ovelimab  5679
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