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Theorem oooeqim2 25156
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 5505 . . . . . 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 5505 . . . . . 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 5024 . . 3  |-  ( ( F " X )  =  ( F " Y )  ->  ( `' F " ( F
" X ) )  =  ( `' F " ( F " Y
) ) )
8 eqtr 2313 . . . . . . . 8  |-  ( ( X  =  ( `' F " ( F
" X ) )  /\  ( `' F " ( F " X
) )  =  ( `' F " ( F
" Y ) ) )  ->  X  =  ( `' F " ( F
" Y ) ) )
9 eqtr 2313 . . . . . . . . 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 2299 . . . 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 5024 . 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 1632    C_ wss 3165   `'ccnv 4704   "cima 4708   -1-1->wf1 5268
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278
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