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Theorem ssimaexg 5745
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 5153 . . . . . 6  |-  ( y  =  A  ->  ( F " y )  =  ( F " A
) )
21sseq2d 3372 . . . . 5  |-  ( y  =  A  ->  ( B  C_  ( F "
y )  <->  B  C_  ( F " A ) ) )
32anbi2d 696 . . . 4  |-  ( y  =  A  ->  (
( Fun  F  /\  B  C_  ( F "
y ) )  <->  ( Fun  F  /\  B  C_  ( F " A ) ) ) )
4 sseq2 3366 . . . . . 6  |-  ( y  =  A  ->  (
x  C_  y  <->  x  C_  A
) )
54anbi1d 697 . . . . 5  |-  ( y  =  A  ->  (
( x  C_  y  /\  B  =  ( F " x ) )  <-> 
( x  C_  A  /\  B  =  ( F " x ) ) ) )
65exbidv 1679 . . . 4  |-  ( y  =  A  ->  ( E. x ( x  C_  y  /\  B  =  ( F " x ) )  <->  E. x ( x 
C_  A  /\  B  =  ( F "
x ) ) ) )
73, 6imbi12d 320 . . 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 2965 . . . 4  |-  y  e. 
_V
98ssimaex 5744 . . 3  |-  ( ( Fun  F  /\  B  C_  ( F " y
) )  ->  E. x
( x  C_  y  /\  B  =  ( F " x ) ) )
107, 9vtoclg 3019 . 2  |-  ( A  e.  C  ->  (
( Fun  F  /\  B  C_  ( F " A ) )  ->  E. x ( x  C_  A  /\  B  =  ( F " x ) ) ) )
11103impib 1178 1  |-  ( ( A  e.  C  /\  Fun  F  /\  B  C_  ( F " A ) )  ->  E. x
( x  C_  A  /\  B  =  ( F " x ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 958    = wceq 1362   E.wex 1589    e. wcel 1755    C_ wss 3316   "cima 4830   Fun wfun 5400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pr 4519
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-sbc 3176  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-fv 5414
This theorem is referenced by:  tgrest  18605  cmpfi  18853
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