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Theorem f1imass 5441
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 495 . . . . . . 7  |-  ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A
) )  /\  ( F " C )  C_  ( F " D ) )  ->  C  C_  A
)
21sseld 2972 . . . . . 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 490 . . . . . . . . 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 2972 . . . . . . . 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 493 . . . . . . . . 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 107 . . . . . . . . 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 980 . . . . . . . . . 10  |-  ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A
) )  /\  ( F " C )  C_  ( F " D )  /\  a  e.  A
)  ->  C  C_  A
)
873expa 1115 . . . . . . . . 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 5440 . . . . . . . . 9  |-  ( ( F : A -1-1-> B  /\  a  e.  A  /\  C  C_  A )  ->  ( ( F `
 a )  e.  ( F " C
)  <->  a  e.  C
) )
105, 6, 8, 9syl3anc 1146 . . . . . . . 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 981 . . . . . . . . . 10  |-  ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A
) )  /\  ( F " C )  C_  ( F " D )  /\  a  e.  A
)  ->  D  C_  A
)
12113expa 1115 . . . . . . . . 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 5440 . . . . . . . . 9  |-  ( ( F : A -1-1-> B  /\  a  e.  A  /\  D  C_  A )  ->  ( ( F `
 a )  e.  ( F " D
)  <->  a  e.  D
) )
145, 6, 12, 13syl3anc 1146 . . . . . . . 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 195 . . . . . . 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 112 . . . . . 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 48 . . . 4  |-  ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A
) )  /\  ( F " C )  C_  ( F " D ) )  ->  ( a  e.  C  ->  a  e.  D ) )
1918ssrdv 2979 . . 3  |-  ( ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A
) )  /\  ( F " C )  C_  ( F " D ) )  ->  C  C_  D
)
2019ex 112 . 2  |-  ( ( F : A -1-1-> B  /\  ( C  C_  A  /\  D  C_  A ) )  ->  ( ( F " C )  C_  ( F " D )  ->  C  C_  D
) )
21 imass2 4729 . 2  |-  ( C 
C_  D  ->  ( F " C )  C_  ( F " D ) )
2220, 21impbid1 134 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 101    <-> wb 102    e. wcel 1409    C_ wss 2945   "cima 4376   -1-1->wf1 4927   ` cfv 4930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fv 4938
This theorem is referenced by:  f1imaeq  5442  f1imapss  5443
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