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Theorem f1imass 5950
Description: Taking images under a one-to-one function preserves subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.)
Assertion
Ref Expression
f1imass  |-  ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A ) )  ->  ( ( F " C )  C_  ( F " D )  <-> 
C  C_  D )
)

Proof of Theorem f1imass
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 simplrl 737 . . . . . . 7  |-  ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A
) )  /\  ( F " C )  C_  ( F " D ) )  ->  C  C_  A
)
21sseld 3291 . . . . . 6  |-  ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A
) )  /\  ( F " C )  C_  ( F " D ) )  ->  ( a  e.  C  ->  a  e.  A ) )
3 simplr 732 . . . . . . . . 9  |-  ( ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A ) )  /\  ( F " C ) 
C_  ( F " D ) )  /\  a  e.  A )  ->  ( F " C
)  C_  ( F " D ) )
43sseld 3291 . . . . . . . 8  |-  ( ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A ) )  /\  ( F " C ) 
C_  ( F " D ) )  /\  a  e.  A )  ->  ( ( F `  a )  e.  ( F " C )  ->  ( F `  a )  e.  ( F " D ) ) )
5 simplll 735 . . . . . . . . 9  |-  ( ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A ) )  /\  ( F " C ) 
C_  ( F " D ) )  /\  a  e.  A )  ->  F : A -1-1-> B
)
6 simpr 448 . . . . . . . . 9  |-  ( ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A ) )  /\  ( F " C ) 
C_  ( F " D ) )  /\  a  e.  A )  ->  a  e.  A )
7 simp1rl 1022 . . . . . . . . . 10  |-  ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A
) )  /\  ( F " C )  C_  ( F " D )  /\  a  e.  A
)  ->  C  C_  A
)
873expa 1153 . . . . . . . . 9  |-  ( ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A ) )  /\  ( F " C ) 
C_  ( F " D ) )  /\  a  e.  A )  ->  C  C_  A )
9 f1elima 5949 . . . . . . . . 9  |-  ( ( F : A -1-1-> B  /\  a  e.  A  /\  C  C_  A )  ->  ( ( F `
 a )  e.  ( F " C
)  <->  a  e.  C
) )
105, 6, 8, 9syl3anc 1184 . . . . . . . 8  |-  ( ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A ) )  /\  ( F " C ) 
C_  ( F " D ) )  /\  a  e.  A )  ->  ( ( F `  a )  e.  ( F " C )  <-> 
a  e.  C ) )
11 simp1rr 1023 . . . . . . . . . 10  |-  ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A
) )  /\  ( F " C )  C_  ( F " D )  /\  a  e.  A
)  ->  D  C_  A
)
12113expa 1153 . . . . . . . . 9  |-  ( ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A ) )  /\  ( F " C ) 
C_  ( F " D ) )  /\  a  e.  A )  ->  D  C_  A )
13 f1elima 5949 . . . . . . . . 9  |-  ( ( F : A -1-1-> B  /\  a  e.  A  /\  D  C_  A )  ->  ( ( F `
 a )  e.  ( F " D
)  <->  a  e.  D
) )
145, 6, 12, 13syl3anc 1184 . . . . . . . 8  |-  ( ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A ) )  /\  ( F " C ) 
C_  ( F " D ) )  /\  a  e.  A )  ->  ( ( F `  a )  e.  ( F " D )  <-> 
a  e.  D ) )
154, 10, 143imtr3d 259 . . . . . . 7  |-  ( ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A ) )  /\  ( F " C ) 
C_  ( F " D ) )  /\  a  e.  A )  ->  ( a  e.  C  ->  a  e.  D ) )
1615ex 424 . . . . . 6  |-  ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A
) )  /\  ( F " C )  C_  ( F " D ) )  ->  ( a  e.  A  ->  ( a  e.  C  ->  a  e.  D ) ) )
172, 16syld 42 . . . . 5  |-  ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A
) )  /\  ( F " C )  C_  ( F " D ) )  ->  ( a  e.  C  ->  ( a  e.  C  ->  a  e.  D ) ) )
1817pm2.43d 46 . . . 4  |-  ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A
) )  /\  ( F " C )  C_  ( F " D ) )  ->  ( a  e.  C  ->  a  e.  D ) )
1918ssrdv 3298 . . 3  |-  ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A
) )  /\  ( F " C )  C_  ( F " D ) )  ->  C  C_  D
)
2019ex 424 . 2  |-  ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A ) )  ->  ( ( F " C )  C_  ( F " D )  ->  C  C_  D
) )
21 imass2 5181 . 2  |-  ( C 
C_  D  ->  ( F " C )  C_  ( F " D ) )
2220, 21impbid1 195 1  |-  ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A ) )  ->  ( ( F " C )  C_  ( F " D )  <-> 
C  C_  D )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1717    C_ wss 3264   "cima 4822   -1-1->wf1 5392   ` cfv 5395
This theorem is referenced by:  f1imaeq  5951  f1imapss  5952  enfin2i  8135  tsmsf1o  18096
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fv 5403
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