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Theorem f1elima 5515
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 5183 . . . 4  |-  ( F : A -1-1-> B  ->  F  Fn  A )
2 fvelimab 5325 . . . 4  |-  ( ( F  Fn  A  /\  Y  C_  A )  -> 
( ( F `  X )  e.  ( F " Y )  <->  E. z  e.  Y  ( F `  z )  =  ( F `  X ) ) )
31, 2sylan 277 . . 3  |-  ( ( F : A -1-1-> B  /\  Y  C_  A )  ->  ( ( F `
 X )  e.  ( F " Y
)  <->  E. z  e.  Y  ( F `  z )  =  ( F `  X ) ) )
433adant2 960 . 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 3008 . . . . . . . 8  |-  ( Y 
C_  A  ->  (
z  e.  Y  -> 
z  e.  A ) )
65impac 373 . . . . . . 7  |-  ( ( Y  C_  A  /\  z  e.  Y )  ->  ( z  e.  A  /\  z  e.  Y
) )
7 f1fveq 5514 . . . . . . . . . . . 12  |-  ( ( F : A -1-1-> B  /\  ( z  e.  A  /\  X  e.  A
) )  ->  (
( F `  z
)  =  ( F `
 X )  <->  z  =  X ) )
87ancom2s 531 . . . . . . . . . . 11  |-  ( ( F : A -1-1-> B  /\  ( X  e.  A  /\  z  e.  A
) )  ->  (
( F `  z
)  =  ( F `
 X )  <->  z  =  X ) )
98biimpd 142 . . . . . . . . . 10  |-  ( ( F : A -1-1-> B  /\  ( X  e.  A  /\  z  e.  A
) )  ->  (
( F `  z
)  =  ( F `
 X )  -> 
z  =  X ) )
109anassrs 392 . . . . . . . . 9  |-  ( ( ( F : A -1-1-> B  /\  X  e.  A
)  /\  z  e.  A )  ->  (
( F `  z
)  =  ( F `
 X )  -> 
z  =  X ) )
11 eleq1 2147 . . . . . . . . . 10  |-  ( z  =  X  ->  (
z  e.  Y  <->  X  e.  Y ) )
1211biimpcd 157 . . . . . . . . 9  |-  ( z  e.  Y  ->  (
z  =  X  ->  X  e.  Y )
)
1310, 12sylan9 401 . . . . . . . 8  |-  ( ( ( ( F : A -1-1-> B  /\  X  e.  A )  /\  z  e.  A )  /\  z  e.  Y )  ->  (
( F `  z
)  =  ( F `
 X )  ->  X  e.  Y )
)
1413anasss 391 . . . . . . 7  |-  ( ( ( F : A -1-1-> B  /\  X  e.  A
)  /\  ( z  e.  A  /\  z  e.  Y ) )  -> 
( ( F `  z )  =  ( F `  X )  ->  X  e.  Y
) )
156, 14sylan2 280 . . . . . 6  |-  ( ( ( F : A -1-1-> B  /\  X  e.  A
)  /\  ( Y  C_  A  /\  z  e.  Y ) )  -> 
( ( F `  z )  =  ( F `  X )  ->  X  e.  Y
) )
1615anassrs 392 . . . . 5  |-  ( ( ( ( F : A -1-1-> B  /\  X  e.  A )  /\  Y  C_  A )  /\  z  e.  Y )  ->  (
( F `  z
)  =  ( F `
 X )  ->  X  e.  Y )
)
1716rexlimdva 2485 . . . 4  |-  ( ( ( F : A -1-1-> B  /\  X  e.  A
)  /\  Y  C_  A
)  ->  ( E. z  e.  Y  ( F `  z )  =  ( F `  X )  ->  X  e.  Y ) )
18173impa 1136 . . 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 2085 . . . 4  |-  ( F `
 X )  =  ( F `  X
)
20 fveq2 5270 . . . . . 6  |-  ( z  =  X  ->  ( F `  z )  =  ( F `  X ) )
2120eqeq1d 2093 . . . . 5  |-  ( z  =  X  ->  (
( F `  z
)  =  ( F `
 X )  <->  ( F `  X )  =  ( F `  X ) ) )
2221rspcev 2715 . . . 4  |-  ( ( X  e.  Y  /\  ( F `  X )  =  ( F `  X ) )  ->  E. z  e.  Y  ( F `  z )  =  ( F `  X ) )
2319, 22mpan2 416 . . 3  |-  ( X  e.  Y  ->  E. z  e.  Y  ( F `  z )  =  ( F `  X ) )
2418, 23impbid1 140 . 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 186 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 102    <-> wb 103    /\ w3a 922    = wceq 1287    e. wcel 1436   E.wrex 2356    C_ wss 2988   "cima 4416    Fn wfn 4978   -1-1->wf1 4980   ` cfv 4983
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 3934  ax-pow 3986  ax-pr 4012
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 3639  df-br 3823  df-opab 3877  df-id 4096  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424  df-res 4425  df-ima 4426  df-iota 4948  df-fun 4985  df-fn 4986  df-f 4987  df-f1 4988  df-fv 4991
This theorem is referenced by:  f1imass  5516  iseqf1olemnab  9841
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