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Theorem f1elima 5787
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 5438 . . . 4  |-  ( F : A -1-1-> B  ->  F  Fn  A )
2 fvelimab 5578 . . . 4  |-  ( ( F  Fn  A  /\  Y  C_  A )  -> 
( ( F `  X )  e.  ( F " Y )  <->  E. z  e.  Y  ( F `  z )  =  ( F `  X ) ) )
31, 2sylan 457 . . 3  |-  ( ( F : A -1-1-> B  /\  Y  C_  A )  ->  ( ( F `
 X )  e.  ( F " Y
)  <->  E. z  e.  Y  ( F `  z )  =  ( F `  X ) ) )
433adant2 974 . 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 3174 . . . . . . . 8  |-  ( Y 
C_  A  ->  (
z  e.  Y  -> 
z  e.  A ) )
65impac 604 . . . . . . 7  |-  ( ( Y  C_  A  /\  z  e.  Y )  ->  ( z  e.  A  /\  z  e.  Y
) )
7 f1fveq 5786 . . . . . . . . . . . 12  |-  ( ( F : A -1-1-> B  /\  ( z  e.  A  /\  X  e.  A
) )  ->  (
( F `  z
)  =  ( F `
 X )  <->  z  =  X ) )
87ancom2s 777 . . . . . . . . . . 11  |-  ( ( F : A -1-1-> B  /\  ( X  e.  A  /\  z  e.  A
) )  ->  (
( F `  z
)  =  ( F `
 X )  <->  z  =  X ) )
98biimpd 198 . . . . . . . . . 10  |-  ( ( F : A -1-1-> B  /\  ( X  e.  A  /\  z  e.  A
) )  ->  (
( F `  z
)  =  ( F `
 X )  -> 
z  =  X ) )
109anassrs 629 . . . . . . . . 9  |-  ( ( ( F : A -1-1-> B  /\  X  e.  A
)  /\  z  e.  A )  ->  (
( F `  z
)  =  ( F `
 X )  -> 
z  =  X ) )
11 eleq1 2343 . . . . . . . . . 10  |-  ( z  =  X  ->  (
z  e.  Y  <->  X  e.  Y ) )
1211biimpcd 215 . . . . . . . . 9  |-  ( z  e.  Y  ->  (
z  =  X  ->  X  e.  Y )
)
1310, 12sylan9 638 . . . . . . . 8  |-  ( ( ( ( F : A -1-1-> B  /\  X  e.  A )  /\  z  e.  A )  /\  z  e.  Y )  ->  (
( F `  z
)  =  ( F `
 X )  ->  X  e.  Y )
)
1413anasss 628 . . . . . . 7  |-  ( ( ( F : A -1-1-> B  /\  X  e.  A
)  /\  ( z  e.  A  /\  z  e.  Y ) )  -> 
( ( F `  z )  =  ( F `  X )  ->  X  e.  Y
) )
156, 14sylan2 460 . . . . . 6  |-  ( ( ( F : A -1-1-> B  /\  X  e.  A
)  /\  ( Y  C_  A  /\  z  e.  Y ) )  -> 
( ( F `  z )  =  ( F `  X )  ->  X  e.  Y
) )
1615anassrs 629 . . . . 5  |-  ( ( ( ( F : A -1-1-> B  /\  X  e.  A )  /\  Y  C_  A )  /\  z  e.  Y )  ->  (
( F `  z
)  =  ( F `
 X )  ->  X  e.  Y )
)
1716rexlimdva 2667 . . . 4  |-  ( ( ( F : A -1-1-> B  /\  X  e.  A
)  /\  Y  C_  A
)  ->  ( E. z  e.  Y  ( F `  z )  =  ( F `  X )  ->  X  e.  Y ) )
18173impa 1146 . . 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 2283 . . . 4  |-  ( F `
 X )  =  ( F `  X
)
20 fveq2 5525 . . . . . 6  |-  ( z  =  X  ->  ( F `  z )  =  ( F `  X ) )
2120eqeq1d 2291 . . . . 5  |-  ( z  =  X  ->  (
( F `  z
)  =  ( F `
 X )  <->  ( F `  X )  =  ( F `  X ) ) )
2221rspcev 2884 . . . 4  |-  ( ( X  e.  Y  /\  ( F `  X )  =  ( F `  X ) )  ->  E. z  e.  Y  ( F `  z )  =  ( F `  X ) )
2319, 22mpan2 652 . . 3  |-  ( X  e.  Y  ->  E. z  e.  Y  ( F `  z )  =  ( F `  X ) )
2418, 23impbid1 194 . 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 244 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    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544    C_ wss 3152   "cima 4692    Fn wfn 5250   -1-1->wf1 5252   ` cfv 5255
This theorem is referenced by:  f1imass  5788  domunfican  7129  acndom2  7681  hashf1lem1  11393  gsumzaddlem  15203  eupath2lem3  23903  axcontlem10  24601  f1elimaOLD  26396  ismtyima  26527  lindfmm  27297  f1omvdconj  27389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fv 5263
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