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Theorem ssimaexg 5585
Description: The existence of a subimage. (Contributed by FL, 15-Apr-2007.)
Assertion
Ref Expression
ssimaexg  |-  ( ( A  e.  C  /\  Fun  F  /\  B  C_  ( F " A ) )  ->  E. x
( x  C_  A  /\  B  =  ( F " x ) ) )
Distinct variable groups:    x, A    x, B    x, F
Allowed substitution hint:    C( x)

Proof of Theorem ssimaexg
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 imaeq2 5008 . . . . . 6  |-  ( y  =  A  ->  ( F " y )  =  ( F " A
) )
21sseq2d 3206 . . . . 5  |-  ( y  =  A  ->  ( B  C_  ( F "
y )  <->  B  C_  ( F " A ) ) )
32anbi2d 684 . . . 4  |-  ( y  =  A  ->  (
( Fun  F  /\  B  C_  ( F "
y ) )  <->  ( Fun  F  /\  B  C_  ( F " A ) ) ) )
4 sseq2 3200 . . . . . 6  |-  ( y  =  A  ->  (
x  C_  y  <->  x  C_  A
) )
54anbi1d 685 . . . . 5  |-  ( y  =  A  ->  (
( x  C_  y  /\  B  =  ( F " x ) )  <-> 
( x  C_  A  /\  B  =  ( F " x ) ) ) )
65exbidv 1612 . . . 4  |-  ( y  =  A  ->  ( E. x ( x  C_  y  /\  B  =  ( F " x ) )  <->  E. x ( x 
C_  A  /\  B  =  ( F "
x ) ) ) )
73, 6imbi12d 311 . . 3  |-  ( y  =  A  ->  (
( ( Fun  F  /\  B  C_  ( F
" y ) )  ->  E. x ( x 
C_  y  /\  B  =  ( F "
x ) ) )  <-> 
( ( Fun  F  /\  B  C_  ( F
" A ) )  ->  E. x ( x 
C_  A  /\  B  =  ( F "
x ) ) ) ) )
8 vex 2791 . . . 4  |-  y  e. 
_V
98ssimaex 5584 . . 3  |-  ( ( Fun  F  /\  B  C_  ( F " y
) )  ->  E. x
( x  C_  y  /\  B  =  ( F " x ) ) )
107, 9vtoclg 2843 . 2  |-  ( A  e.  C  ->  (
( Fun  F  /\  B  C_  ( F " A ) )  ->  E. x ( x  C_  A  /\  B  =  ( F " x ) ) ) )
11103impib 1149 1  |-  ( ( A  e.  C  /\  Fun  F  /\  B  C_  ( F " A ) )  ->  E. x
( x  C_  A  /\  B  =  ( F " x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684    C_ wss 3152   "cima 4692   Fun wfun 5249
This theorem is referenced by:  tgrest  16890  cmpfi  17135
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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