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Theorem f1elima 5768
Description: Membership in the image of a 1-1 map. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
f1elima  |-  ( ( F : A -1-1-> B  /\  X  e.  A  /\  Y  C_  A )  ->  ( ( F `
 X )  e.  ( F " Y
)  <->  X  e.  Y
) )

Proof of Theorem f1elima
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 f1fn 5419 . . . 4  |-  ( F : A -1-1-> B  ->  F  Fn  A )
2 fvelimab 5568 . . . 4  |-  ( ( F  Fn  A  /\  Y  C_  A )  -> 
( ( F `  X )  e.  ( F " Y )  <->  E. z  e.  Y  ( F `  z )  =  ( F `  X ) ) )
31, 2sylan 283 . . 3  |-  ( ( F : A -1-1-> B  /\  Y  C_  A )  ->  ( ( F `
 X )  e.  ( F " Y
)  <->  E. z  e.  Y  ( F `  z )  =  ( F `  X ) ) )
433adant2 1016 . 2  |-  ( ( F : A -1-1-> B  /\  X  e.  A  /\  Y  C_  A )  ->  ( ( F `
 X )  e.  ( F " Y
)  <->  E. z  e.  Y  ( F `  z )  =  ( F `  X ) ) )
5 ssel 3149 . . . . . . . 8  |-  ( Y 
C_  A  ->  (
z  e.  Y  -> 
z  e.  A ) )
65impac 381 . . . . . . 7  |-  ( ( Y  C_  A  /\  z  e.  Y )  ->  ( z  e.  A  /\  z  e.  Y
) )
7 f1fveq 5767 . . . . . . . . . . . 12  |-  ( ( F : A -1-1-> B  /\  ( z  e.  A  /\  X  e.  A
) )  ->  (
( F `  z
)  =  ( F `
 X )  <->  z  =  X ) )
87ancom2s 566 . . . . . . . . . . 11  |-  ( ( F : A -1-1-> B  /\  ( X  e.  A  /\  z  e.  A
) )  ->  (
( F `  z
)  =  ( F `
 X )  <->  z  =  X ) )
98biimpd 144 . . . . . . . . . 10  |-  ( ( F : A -1-1-> B  /\  ( X  e.  A  /\  z  e.  A
) )  ->  (
( F `  z
)  =  ( F `
 X )  -> 
z  =  X ) )
109anassrs 400 . . . . . . . . 9  |-  ( ( ( F : A -1-1-> B  /\  X  e.  A
)  /\  z  e.  A )  ->  (
( F `  z
)  =  ( F `
 X )  -> 
z  =  X ) )
11 eleq1 2240 . . . . . . . . . 10  |-  ( z  =  X  ->  (
z  e.  Y  <->  X  e.  Y ) )
1211biimpcd 159 . . . . . . . . 9  |-  ( z  e.  Y  ->  (
z  =  X  ->  X  e.  Y )
)
1310, 12sylan9 409 . . . . . . . 8  |-  ( ( ( ( F : A -1-1-> B  /\  X  e.  A )  /\  z  e.  A )  /\  z  e.  Y )  ->  (
( F `  z
)  =  ( F `
 X )  ->  X  e.  Y )
)
1413anasss 399 . . . . . . 7  |-  ( ( ( F : A -1-1-> B  /\  X  e.  A
)  /\  ( z  e.  A  /\  z  e.  Y ) )  -> 
( ( F `  z )  =  ( F `  X )  ->  X  e.  Y
) )
156, 14sylan2 286 . . . . . 6  |-  ( ( ( F : A -1-1-> B  /\  X  e.  A
)  /\  ( Y  C_  A  /\  z  e.  Y ) )  -> 
( ( F `  z )  =  ( F `  X )  ->  X  e.  Y
) )
1615anassrs 400 . . . . 5  |-  ( ( ( ( F : A -1-1-> B  /\  X  e.  A )  /\  Y  C_  A )  /\  z  e.  Y )  ->  (
( F `  z
)  =  ( F `
 X )  ->  X  e.  Y )
)
1716rexlimdva 2594 . . . 4  |-  ( ( ( F : A -1-1-> B  /\  X  e.  A
)  /\  Y  C_  A
)  ->  ( E. z  e.  Y  ( F `  z )  =  ( F `  X )  ->  X  e.  Y ) )
18173impa 1194 . . 3  |-  ( ( F : A -1-1-> B  /\  X  e.  A  /\  Y  C_  A )  ->  ( E. z  e.  Y  ( F `  z )  =  ( F `  X )  ->  X  e.  Y
) )
19 eqid 2177 . . . 4  |-  ( F `
 X )  =  ( F `  X
)
20 fveq2 5511 . . . . . 6  |-  ( z  =  X  ->  ( F `  z )  =  ( F `  X ) )
2120eqeq1d 2186 . . . . 5  |-  ( z  =  X  ->  (
( F `  z
)  =  ( F `
 X )  <->  ( F `  X )  =  ( F `  X ) ) )
2221rspcev 2841 . . . 4  |-  ( ( X  e.  Y  /\  ( F `  X )  =  ( F `  X ) )  ->  E. z  e.  Y  ( F `  z )  =  ( F `  X ) )
2319, 22mpan2 425 . . 3  |-  ( X  e.  Y  ->  E. z  e.  Y  ( F `  z )  =  ( F `  X ) )
2418, 23impbid1 142 . 2  |-  ( ( F : A -1-1-> B  /\  X  e.  A  /\  Y  C_  A )  ->  ( E. z  e.  Y  ( F `  z )  =  ( F `  X )  <-> 
X  e.  Y ) )
254, 24bitrd 188 1  |-  ( ( F : A -1-1-> B  /\  X  e.  A  /\  Y  C_  A )  ->  ( ( F `
 X )  e.  ( F " Y
)  <->  X  e.  Y
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   E.wrex 2456    C_ wss 3129   "cima 4626    Fn wfn 5207   -1-1->wf1 5209   ` cfv 5212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fv 5220
This theorem is referenced by:  f1imass  5769  iseqf1olemnab  10471  fprodssdc  11579  ctinfom  12409  ssnnctlemct  12427
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