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Theorem f1imass 5514
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 502 . . . . . . 7  |-  ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A
) )  /\  ( F " C )  C_  ( F " D ) )  ->  C  C_  A
)
21sseld 3013 . . . . . 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 497 . . . . . . . . 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 3013 . . . . . . . 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 500 . . . . . . . . 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 108 . . . . . . . . 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 1006 . . . . . . . . . 10  |-  ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A
) )  /\  ( F " C )  C_  ( F " D )  /\  a  e.  A
)  ->  C  C_  A
)
873expa 1141 . . . . . . . . 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 5513 . . . . . . . . 9  |-  ( ( F : A -1-1-> B  /\  a  e.  A  /\  C  C_  A )  ->  ( ( F `
 a )  e.  ( F " C
)  <->  a  e.  C
) )
105, 6, 8, 9syl3anc 1172 . . . . . . . 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 1007 . . . . . . . . . 10  |-  ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A
) )  /\  ( F " C )  C_  ( F " D )  /\  a  e.  A
)  ->  D  C_  A
)
12113expa 1141 . . . . . . . . 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 5513 . . . . . . . . 9  |-  ( ( F : A -1-1-> B  /\  a  e.  A  /\  D  C_  A )  ->  ( ( F `
 a )  e.  ( F " D
)  <->  a  e.  D
) )
145, 6, 12, 13syl3anc 1172 . . . . . . . 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 200 . . . . . . 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 113 . . . . . 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 44 . . . . 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 49 . . . 4  |-  ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A
) )  /\  ( F " C )  C_  ( F " D ) )  ->  ( a  e.  C  ->  a  e.  D ) )
1918ssrdv 3020 . . 3  |-  ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A
) )  /\  ( F " C )  C_  ( F " D ) )  ->  C  C_  D
)
2019ex 113 . 2  |-  ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A ) )  ->  ( ( F " C )  C_  ( F " D )  ->  C  C_  D
) )
21 imass2 4775 . 2  |-  ( C 
C_  D  ->  ( F " C )  C_  ( F " D ) )
2220, 21impbid1 140 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    /\ wa 102    <-> wb 103    e. wcel 1436    C_ wss 2988   "cima 4414   -1-1->wf1 4978   ` cfv 4981
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-sbc 2830  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-br 3821  df-opab 3875  df-id 4094  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-res 4423  df-ima 4424  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-f1 4986  df-fv 4989
This theorem is referenced by:  f1imaeq  5515
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