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Theorem ssimaexg 5939
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 5343 . . . . . 6  |-  ( y  =  A  ->  ( F " y )  =  ( F " A
) )
21sseq2d 3527 . . . . 5  |-  ( y  =  A  ->  ( B  C_  ( F "
y )  <->  B  C_  ( F " A ) ) )
32anbi2d 703 . . . 4  |-  ( y  =  A  ->  (
( Fun  F  /\  B  C_  ( F "
y ) )  <->  ( Fun  F  /\  B  C_  ( F " A ) ) ) )
4 sseq2 3521 . . . . . 6  |-  ( y  =  A  ->  (
x  C_  y  <->  x  C_  A
) )
54anbi1d 704 . . . . 5  |-  ( y  =  A  ->  (
( x  C_  y  /\  B  =  ( F " x ) )  <-> 
( x  C_  A  /\  B  =  ( F " x ) ) ) )
65exbidv 1715 . . . 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 3112 . . . 4  |-  y  e. 
_V
98ssimaex 5938 . . 3  |-  ( ( Fun  F  /\  B  C_  ( F " y
) )  ->  E. x
( x  C_  y  /\  B  =  ( F " x ) ) )
107, 9vtoclg 3167 . 2  |-  ( A  e.  C  ->  (
( Fun  F  /\  B  C_  ( F " A ) )  ->  E. x ( x  C_  A  /\  B  =  ( F " x ) ) ) )
11103impib 1194 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 973    = wceq 1395   E.wex 1613    e. wcel 1819    C_ wss 3471   "cima 5011   Fun wfun 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602
This theorem is referenced by:  tgrest  19787  cmpfi  20035
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