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Theorem ssimaexg 5790
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 5200 . . . . . 6  |-  ( y  =  A  ->  ( F " y )  =  ( F " A
) )
21sseq2d 3377 . . . . 5  |-  ( y  =  A  ->  ( B  C_  ( F "
y )  <->  B  C_  ( F " A ) ) )
32anbi2d 686 . . . 4  |-  ( y  =  A  ->  (
( Fun  F  /\  B  C_  ( F "
y ) )  <->  ( Fun  F  /\  B  C_  ( F " A ) ) ) )
4 sseq2 3371 . . . . . 6  |-  ( y  =  A  ->  (
x  C_  y  <->  x  C_  A
) )
54anbi1d 687 . . . . 5  |-  ( y  =  A  ->  (
( x  C_  y  /\  B  =  ( F " x ) )  <-> 
( x  C_  A  /\  B  =  ( F " x ) ) ) )
65exbidv 1637 . . . 4  |-  ( y  =  A  ->  ( E. x ( x  C_  y  /\  B  =  ( F " x ) )  <->  E. x ( x 
C_  A  /\  B  =  ( F "
x ) ) ) )
73, 6imbi12d 313 . . 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 2960 . . . 4  |-  y  e. 
_V
98ssimaex 5789 . . 3  |-  ( ( Fun  F  /\  B  C_  ( F " y
) )  ->  E. x
( x  C_  y  /\  B  =  ( F " x ) ) )
107, 9vtoclg 3012 . 2  |-  ( A  e.  C  ->  (
( Fun  F  /\  B  C_  ( F " A ) )  ->  E. x ( x  C_  A  /\  B  =  ( F " x ) ) ) )
11103impib 1152 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 360    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1726    C_ wss 3321   "cima 4882   Fun wfun 5449
This theorem is referenced by:  tgrest  17224  cmpfi  17472
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-fv 5463
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