ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ssimaexg Unicode version

Theorem ssimaexg 5641
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 5018 . . . . . 6  |-  ( y  =  A  ->  ( F " y )  =  ( F " A
) )
21sseq2d 3223 . . . . 5  |-  ( y  =  A  ->  ( B  C_  ( F "
y )  <->  B  C_  ( F " A ) ) )
32anbi2d 464 . . . 4  |-  ( y  =  A  ->  (
( Fun  F  /\  B  C_  ( F "
y ) )  <->  ( Fun  F  /\  B  C_  ( F " A ) ) ) )
4 sseq2 3217 . . . . . 6  |-  ( y  =  A  ->  (
x  C_  y  <->  x  C_  A
) )
54anbi1d 465 . . . . 5  |-  ( y  =  A  ->  (
( x  C_  y  /\  B  =  ( F " x ) )  <-> 
( x  C_  A  /\  B  =  ( F " x ) ) ) )
65exbidv 1848 . . . 4  |-  ( y  =  A  ->  ( E. x ( x  C_  y  /\  B  =  ( F " x ) )  <->  E. x ( x 
C_  A  /\  B  =  ( F "
x ) ) ) )
73, 6imbi12d 234 . . 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 2775 . . . 4  |-  y  e. 
_V
98ssimaex 5640 . . 3  |-  ( ( Fun  F  /\  B  C_  ( F " y
) )  ->  E. x
( x  C_  y  /\  B  =  ( F " x ) ) )
107, 9vtoclg 2833 . 2  |-  ( A  e.  C  ->  (
( Fun  F  /\  B  C_  ( F " A ) )  ->  E. x ( x  C_  A  /\  B  =  ( F " x ) ) ) )
11103impib 1204 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 104    /\ w3a 981    = wceq 1373   E.wex 1515    e. wcel 2176    C_ wss 3166   "cima 4678   Fun wfun 5265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253
This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-fv 5279
This theorem is referenced by:  tgrest  14641
  Copyright terms: Public domain W3C validator