ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  f1imass Unicode version

Theorem f1imass 5413
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 487 . . . . . . 7  |-  ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A
) )  /\  ( F " C )  C_  ( F " D ) )  ->  C  C_  A
)
21sseld 2944 . . . . . 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 482 . . . . . . . . 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 2944 . . . . . . . 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 485 . . . . . . . . 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 103 . . . . . . . . 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 969 . . . . . . . . . 10  |-  ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A
) )  /\  ( F " C )  C_  ( F " D )  /\  a  e.  A
)  ->  C  C_  A
)
873expa 1104 . . . . . . . . 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 5412 . . . . . . . . 9  |-  ( ( F : A -1-1-> B  /\  a  e.  A  /\  C  C_  A )  ->  ( ( F `
 a )  e.  ( F " C
)  <->  a  e.  C
) )
105, 6, 8, 9syl3anc 1135 . . . . . . . 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 970 . . . . . . . . . 10  |-  ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A
) )  /\  ( F " C )  C_  ( F " D )  /\  a  e.  A
)  ->  D  C_  A
)
12113expa 1104 . . . . . . . . 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 5412 . . . . . . . . 9  |-  ( ( F : A -1-1-> B  /\  a  e.  A  /\  D  C_  A )  ->  ( ( F `
 a )  e.  ( F " D
)  <->  a  e.  D
) )
145, 6, 12, 13syl3anc 1135 . . . . . . . 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 191 . . . . . . 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 108 . . . . . 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 40 . . . . 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 44 . . . 4  |-  ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A
) )  /\  ( F " C )  C_  ( F " D ) )  ->  ( a  e.  C  ->  a  e.  D ) )
1918ssrdv 2951 . . 3  |-  ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A
) )  /\  ( F " C )  C_  ( F " D ) )  ->  C  C_  D
)
2019ex 108 . 2  |-  ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A ) )  ->  ( ( F " C )  C_  ( F " D )  ->  C  C_  D
) )
21 imass2 4701 . 2  |-  ( C 
C_  D  ->  ( F " C )  C_  ( F " D ) )
2220, 21impbid1 130 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 97    <-> wb 98    e. wcel 1393    C_ wss 2917   "cima 4348   -1-1->wf1 4899   ` cfv 4902
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fv 4910
This theorem is referenced by:  f1imaeq  5414  f1imapss  5415
  Copyright terms: Public domain W3C validator