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Theorem f1elima 6011
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 5642 . . . 4  |-  ( F : A -1-1-> B  ->  F  Fn  A )
2 fvelimab 5784 . . . 4  |-  ( ( F  Fn  A  /\  Y  C_  A )  -> 
( ( F `  X )  e.  ( F " Y )  <->  E. z  e.  Y  ( F `  z )  =  ( F `  X ) ) )
31, 2sylan 459 . . 3  |-  ( ( F : A -1-1-> B  /\  Y  C_  A )  ->  ( ( F `
 X )  e.  ( F " Y
)  <->  E. z  e.  Y  ( F `  z )  =  ( F `  X ) ) )
433adant2 977 . 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 3344 . . . . . . . 8  |-  ( Y 
C_  A  ->  (
z  e.  Y  -> 
z  e.  A ) )
65impac 606 . . . . . . 7  |-  ( ( Y  C_  A  /\  z  e.  Y )  ->  ( z  e.  A  /\  z  e.  Y
) )
7 f1fveq 6010 . . . . . . . . . . . 12  |-  ( ( F : A -1-1-> B  /\  ( z  e.  A  /\  X  e.  A
) )  ->  (
( F `  z
)  =  ( F `
 X )  <->  z  =  X ) )
87ancom2s 779 . . . . . . . . . . 11  |-  ( ( F : A -1-1-> B  /\  ( X  e.  A  /\  z  e.  A
) )  ->  (
( F `  z
)  =  ( F `
 X )  <->  z  =  X ) )
98biimpd 200 . . . . . . . . . 10  |-  ( ( F : A -1-1-> B  /\  ( X  e.  A  /\  z  e.  A
) )  ->  (
( F `  z
)  =  ( F `
 X )  -> 
z  =  X ) )
109anassrs 631 . . . . . . . . 9  |-  ( ( ( F : A -1-1-> B  /\  X  e.  A
)  /\  z  e.  A )  ->  (
( F `  z
)  =  ( F `
 X )  -> 
z  =  X ) )
11 eleq1 2498 . . . . . . . . . 10  |-  ( z  =  X  ->  (
z  e.  Y  <->  X  e.  Y ) )
1211biimpcd 217 . . . . . . . . 9  |-  ( z  e.  Y  ->  (
z  =  X  ->  X  e.  Y )
)
1310, 12sylan9 640 . . . . . . . 8  |-  ( ( ( ( F : A -1-1-> B  /\  X  e.  A )  /\  z  e.  A )  /\  z  e.  Y )  ->  (
( F `  z
)  =  ( F `
 X )  ->  X  e.  Y )
)
1413anasss 630 . . . . . . 7  |-  ( ( ( F : A -1-1-> B  /\  X  e.  A
)  /\  ( z  e.  A  /\  z  e.  Y ) )  -> 
( ( F `  z )  =  ( F `  X )  ->  X  e.  Y
) )
156, 14sylan2 462 . . . . . 6  |-  ( ( ( F : A -1-1-> B  /\  X  e.  A
)  /\  ( Y  C_  A  /\  z  e.  Y ) )  -> 
( ( F `  z )  =  ( F `  X )  ->  X  e.  Y
) )
1615anassrs 631 . . . . 5  |-  ( ( ( ( F : A -1-1-> B  /\  X  e.  A )  /\  Y  C_  A )  /\  z  e.  Y )  ->  (
( F `  z
)  =  ( F `
 X )  ->  X  e.  Y )
)
1716rexlimdva 2832 . . . 4  |-  ( ( ( F : A -1-1-> B  /\  X  e.  A
)  /\  Y  C_  A
)  ->  ( E. z  e.  Y  ( F `  z )  =  ( F `  X )  ->  X  e.  Y ) )
18173impa 1149 . . 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 2438 . . . 4  |-  ( F `
 X )  =  ( F `  X
)
20 fveq2 5730 . . . . . 6  |-  ( z  =  X  ->  ( F `  z )  =  ( F `  X ) )
2120eqeq1d 2446 . . . . 5  |-  ( z  =  X  ->  (
( F `  z
)  =  ( F `
 X )  <->  ( F `  X )  =  ( F `  X ) ) )
2221rspcev 3054 . . . 4  |-  ( ( X  e.  Y  /\  ( F `  X )  =  ( F `  X ) )  ->  E. z  e.  Y  ( F `  z )  =  ( F `  X ) )
2319, 22mpan2 654 . . 3  |-  ( X  e.  Y  ->  E. z  e.  Y  ( F `  z )  =  ( F `  X ) )
2418, 23impbid1 196 . 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 246 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   E.wrex 2708    C_ wss 3322   "cima 4883    Fn wfn 5451   -1-1->wf1 5453   ` cfv 5456
This theorem is referenced by:  f1imass  6012  domunfican  7381  acndom2  7937  hashf1lem1  11706  gsumzaddlem  15528  eupath2lem3  21703  axcontlem10  25914  ismtyima  26514  lindfmm  27276  f1omvdconj  27368
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fv 5464
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