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Theorem oooeqim2 24465
Description: Symmetrical equality of the images and of their antecedents when the mapping is one to one. (Contributed by FL, 1-Jun-2008.)
Assertion
Ref Expression
oooeqim2  |-  ( ( F : A -1-1-> B  /\  X  C_  A  /\  Y  C_  A )  -> 
( ( F " X )  =  ( F " Y )  <-> 
X  =  Y ) )

Proof of Theorem oooeqim2
StepHypRef Expression
1 f1imacnv 5489 . . . . . 6  |-  ( ( F : A -1-1-> B  /\  X  C_  A )  ->  ( `' F " ( F " X
) )  =  X )
21ex 423 . . . . 5  |-  ( F : A -1-1-> B  -> 
( X  C_  A  ->  ( `' F "
( F " X
) )  =  X ) )
3 f1imacnv 5489 . . . . . 6  |-  ( ( F : A -1-1-> B  /\  Y  C_  A )  ->  ( `' F " ( F " Y
) )  =  Y )
43ex 423 . . . . 5  |-  ( F : A -1-1-> B  -> 
( Y  C_  A  ->  ( `' F "
( F " Y
) )  =  Y ) )
52, 4anim12d 546 . . . 4  |-  ( F : A -1-1-> B  -> 
( ( X  C_  A  /\  Y  C_  A
)  ->  ( ( `' F " ( F
" X ) )  =  X  /\  ( `' F " ( F
" Y ) )  =  Y ) ) )
653impib 1149 . . 3  |-  ( ( F : A -1-1-> B  /\  X  C_  A  /\  Y  C_  A )  -> 
( ( `' F " ( F " X
) )  =  X  /\  ( `' F " ( F " Y
) )  =  Y ) )
7 imaeq2 5008 . . 3  |-  ( ( F " X )  =  ( F " Y )  ->  ( `' F " ( F
" X ) )  =  ( `' F " ( F " Y
) ) )
8 eqtr 2300 . . . . . . . 8  |-  ( ( X  =  ( `' F " ( F
" X ) )  /\  ( `' F " ( F " X
) )  =  ( `' F " ( F
" Y ) ) )  ->  X  =  ( `' F " ( F
" Y ) ) )
9 eqtr 2300 . . . . . . . . 9  |-  ( ( X  =  ( `' F " ( F
" Y ) )  /\  ( `' F " ( F " Y
) )  =  Y )  ->  X  =  Y )
109ex 423 . . . . . . . 8  |-  ( X  =  ( `' F " ( F " Y
) )  ->  (
( `' F "
( F " Y
) )  =  Y  ->  X  =  Y ) )
118, 10syl 15 . . . . . . 7  |-  ( ( X  =  ( `' F " ( F
" X ) )  /\  ( `' F " ( F " X
) )  =  ( `' F " ( F
" Y ) ) )  ->  ( ( `' F " ( F
" Y ) )  =  Y  ->  X  =  Y ) )
1211ex 423 . . . . . 6  |-  ( X  =  ( `' F " ( F " X
) )  ->  (
( `' F "
( F " X
) )  =  ( `' F " ( F
" Y ) )  ->  ( ( `' F " ( F
" Y ) )  =  Y  ->  X  =  Y ) ) )
1312com23 72 . . . . 5  |-  ( X  =  ( `' F " ( F " X
) )  ->  (
( `' F "
( F " Y
) )  =  Y  ->  ( ( `' F " ( F
" X ) )  =  ( `' F " ( F " Y
) )  ->  X  =  Y ) ) )
1413eqcoms 2286 . . . 4  |-  ( ( `' F " ( F
" X ) )  =  X  ->  (
( `' F "
( F " Y
) )  =  Y  ->  ( ( `' F " ( F
" X ) )  =  ( `' F " ( F " Y
) )  ->  X  =  Y ) ) )
1514imp 418 . . 3  |-  ( ( ( `' F "
( F " X
) )  =  X  /\  ( `' F " ( F " Y
) )  =  Y )  ->  ( ( `' F " ( F
" X ) )  =  ( `' F " ( F " Y
) )  ->  X  =  Y ) )
166, 7, 15syl2im 34 . 2  |-  ( ( F : A -1-1-> B  /\  X  C_  A  /\  Y  C_  A )  -> 
( ( F " X )  =  ( F " Y )  ->  X  =  Y ) )
17 imaeq2 5008 . 2  |-  ( X  =  Y  ->  ( F " X )  =  ( F " Y
) )
1816, 17impbid1 194 1  |-  ( ( F : A -1-1-> B  /\  X  C_  A  /\  Y  C_  A )  -> 
( ( F " X )  =  ( F " Y )  <-> 
X  =  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    C_ wss 3152   `'ccnv 4688   "cima 4692   -1-1->wf1 5252
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-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-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-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262
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