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Theorem fvelimab 5550
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 2741 . . . 4  |-  ( C  e.  ( F " B )  ->  C  e.  _V )
21anim2i 340 . . 3  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  C  e.  ( F " B ) )  ->  ( ( F  Fn  A  /\  B  C_  A )  /\  C  e.  _V )
)
3 ssel2 3142 . . . . . . . 8  |-  ( ( B  C_  A  /\  u  e.  B )  ->  u  e.  A )
4 funfvex 5511 . . . . . . . . 9  |-  ( ( Fun  F  /\  u  e.  dom  F )  -> 
( F `  u
)  e.  _V )
54funfni 5296 . . . . . . . 8  |-  ( ( F  Fn  A  /\  u  e.  A )  ->  ( F `  u
)  e.  _V )
63, 5sylan2 284 . . . . . . 7  |-  ( ( F  Fn  A  /\  ( B  C_  A  /\  u  e.  B )
)  ->  ( F `  u )  e.  _V )
76anassrs 398 . . . . . 6  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  u  e.  B
)  ->  ( F `  u )  e.  _V )
8 eleq1 2233 . . . . . 6  |-  ( ( F `  u )  =  C  ->  (
( F `  u
)  e.  _V  <->  C  e.  _V ) )
97, 8syl5ibcom 154 . . . . 5  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  u  e.  B
)  ->  ( ( F `  u )  =  C  ->  C  e. 
_V ) )
109rexlimdva 2587 . . . 4  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( E. u  e.  B  ( F `  u )  =  C  ->  C  e.  _V ) )
1110imdistani 443 . . 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 2233 . . . . . . 7  |-  ( v  =  C  ->  (
v  e.  ( F
" B )  <->  C  e.  ( F " B ) ) )
13 eqeq2 2180 . . . . . . . 8  |-  ( v  =  C  ->  (
( F `  u
)  =  v  <->  ( F `  u )  =  C ) )
1413rexbidv 2471 . . . . . . 7  |-  ( v  =  C  ->  ( E. u  e.  B  ( F `  u )  =  v  <->  E. u  e.  B  ( F `  u )  =  C ) )
1512, 14bibi12d 234 . . . . . 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 229 . . . . 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 5293 . . . . . . . 8  |-  ( F  Fn  A  ->  Fun  F )
1817adantr 274 . . . . . . 7  |-  ( ( F  Fn  A  /\  B  C_  A )  ->  Fun  F )
19 fndm 5295 . . . . . . . . 9  |-  ( F  Fn  A  ->  dom  F  =  A )
2019sseq2d 3177 . . . . . . . 8  |-  ( F  Fn  A  ->  ( B  C_  dom  F  <->  B  C_  A
) )
2120biimpar 295 . . . . . . 7  |-  ( ( F  Fn  A  /\  B  C_  A )  ->  B  C_  dom  F )
22 dfimafn 5543 . . . . . . 7  |-  ( ( Fun  F  /\  B  C_ 
dom  F )  -> 
( F " B
)  =  { v  |  E. u  e.  B  ( F `  u )  =  v } )
2318, 21, 22syl2anc 409 . . . . . 6  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( F " B
)  =  { v  |  E. u  e.  B  ( F `  u )  =  v } )
2423abeq2d 2283 . . . . 5  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( v  e.  ( F " B )  <->  E. u  e.  B  ( F `  u )  =  v ) )
2516, 24vtoclg 2790 . . . 4  |-  ( C  e.  _V  ->  (
( F  Fn  A  /\  B  C_  A )  ->  ( C  e.  ( F " B
)  <->  E. u  e.  B  ( F `  u )  =  C ) ) )
2625impcom 124 . . 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 911 . 2  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( C  e.  ( F " B )  <->  E. u  e.  B  ( F `  u )  =  C ) )
28 fveq2 5494 . . . 4  |-  ( u  =  x  ->  ( F `  u )  =  ( F `  x ) )
2928eqeq1d 2179 . . 3  |-  ( u  =  x  ->  (
( F `  u
)  =  C  <->  ( F `  x )  =  C ) )
3029cbvrexv 2697 . 2  |-  ( E. u  e.  B  ( F `  u )  =  C  <->  E. x  e.  B  ( F `  x )  =  C )
3127, 30bitrdi 195 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 103    <-> wb 104    = wceq 1348    e. wcel 2141   {cab 2156   E.wrex 2449   _Vcvv 2730    C_ wss 3121   dom cdm 4609   "cima 4612   Fun wfun 5190    Fn wfn 5191   ` cfv 5196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-fv 5204
This theorem is referenced by:  ssimaex  5555  foima2  5728  rexima  5731  ralima  5732  f1elima  5749  ovelimab  6000
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