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Theorem f1imass 5668
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 524 . . . . . . 7  |-  ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A
) )  /\  ( F " C )  C_  ( F " D ) )  ->  C  C_  A
)
21sseld 3091 . . . . . 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 519 . . . . . . . . 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 3091 . . . . . . . 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 522 . . . . . . . . 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 109 . . . . . . . . 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 1046 . . . . . . . . . 10  |-  ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A
) )  /\  ( F " C )  C_  ( F " D )  /\  a  e.  A
)  ->  C  C_  A
)
873expa 1181 . . . . . . . . 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 5667 . . . . . . . . 9  |-  ( ( F : A -1-1-> B  /\  a  e.  A  /\  C  C_  A )  ->  ( ( F `
 a )  e.  ( F " C
)  <->  a  e.  C
) )
105, 6, 8, 9syl3anc 1216 . . . . . . . 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 1047 . . . . . . . . . 10  |-  ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A
) )  /\  ( F " C )  C_  ( F " D )  /\  a  e.  A
)  ->  D  C_  A
)
12113expa 1181 . . . . . . . . 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 5667 . . . . . . . . 9  |-  ( ( F : A -1-1-> B  /\  a  e.  A  /\  D  C_  A )  ->  ( ( F `
 a )  e.  ( F " D
)  <->  a  e.  D
) )
145, 6, 12, 13syl3anc 1216 . . . . . . . 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 201 . . . . . . 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 114 . . . . . 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 45 . . . . 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 50 . . . 4  |-  ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A
) )  /\  ( F " C )  C_  ( F " D ) )  ->  ( a  e.  C  ->  a  e.  D ) )
1918ssrdv 3098 . . 3  |-  ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A
) )  /\  ( F " C )  C_  ( F " D ) )  ->  C  C_  D
)
2019ex 114 . 2  |-  ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A ) )  ->  ( ( F " C )  C_  ( F " D )  ->  C  C_  D
) )
21 imass2 4910 . 2  |-  ( C 
C_  D  ->  ( F " C )  C_  ( F " D ) )
2220, 21impbid1 141 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 103    <-> wb 104    e. wcel 1480    C_ wss 3066   "cima 4537   -1-1->wf1 5115   ` cfv 5118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fv 5126
This theorem is referenced by:  f1imaeq  5669
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