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Theorem bnj145 29095
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj145.1  |-  A  e. 
_V
bnj145.2  |-  ( F `
 A )  e. 
_V
Assertion
Ref Expression
bnj145  |-  ( F  Fn  { A }  ->  F  =  { <. A ,  ( F `  A ) >. } )

Proof of Theorem bnj145
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 bnj142 29094 . . . . 5  |-  ( F  Fn  { A }  ->  ( u  e.  F  ->  u  =  <. A , 
( F `  A
) >. ) )
2 df-fn 5458 . . . . . . . 8  |-  ( F  Fn  { A }  <->  ( Fun  F  /\  dom  F  =  { A }
) )
3 bnj145.1 . . . . . . . . . . 11  |-  A  e. 
_V
43snid 3842 . . . . . . . . . 10  |-  A  e. 
{ A }
5 eleq2 2498 . . . . . . . . . 10  |-  ( dom 
F  =  { A }  ->  ( A  e. 
dom  F  <->  A  e.  { A } ) )
64, 5mpbiri 226 . . . . . . . . 9  |-  ( dom 
F  =  { A }  ->  A  e.  dom  F )
76anim2i 554 . . . . . . . 8  |-  ( ( Fun  F  /\  dom  F  =  { A }
)  ->  ( Fun  F  /\  A  e.  dom  F ) )
82, 7sylbi 189 . . . . . . 7  |-  ( F  Fn  { A }  ->  ( Fun  F  /\  A  e.  dom  F ) )
9 funfvop 5843 . . . . . . 7  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  <. A ,  ( F `
 A ) >.  e.  F )
108, 9syl 16 . . . . . 6  |-  ( F  Fn  { A }  -> 
<. A ,  ( F `
 A ) >.  e.  F )
11 eleq1 2497 . . . . . 6  |-  ( u  =  <. A ,  ( F `  A )
>.  ->  ( u  e.  F  <->  <. A ,  ( F `  A )
>.  e.  F ) )
1210, 11syl5ibrcom 215 . . . . 5  |-  ( F  Fn  { A }  ->  ( u  =  <. A ,  ( F `  A ) >.  ->  u  e.  F ) )
131, 12impbid 185 . . . 4  |-  ( F  Fn  { A }  ->  ( u  e.  F  <->  u  =  <. A ,  ( F `  A )
>. ) )
1413alrimiv 1642 . . 3  |-  ( F  Fn  { A }  ->  A. u ( u  e.  F  <->  u  =  <. A ,  ( F `
 A ) >.
) )
15 elsn 3830 . . . . 5  |-  ( u  e.  { <. A , 
( F `  A
) >. }  <->  u  =  <. A ,  ( F `
 A ) >.
)
1615bibi2i 306 . . . 4  |-  ( ( u  e.  F  <->  u  e.  {
<. A ,  ( F `
 A ) >. } )  <->  ( u  e.  F  <->  u  =  <. A ,  ( F `  A ) >. )
)
1716albii 1576 . . 3  |-  ( A. u ( u  e.  F  <->  u  e.  { <. A ,  ( F `  A ) >. } )  <->  A. u ( u  e.  F  <->  u  =  <. A ,  ( F `  A ) >. )
)
1814, 17sylibr 205 . 2  |-  ( F  Fn  { A }  ->  A. u ( u  e.  F  <->  u  e.  {
<. A ,  ( F `
 A ) >. } ) )
19 dfcleq 2431 . 2  |-  ( F  =  { <. A , 
( F `  A
) >. }  <->  A. u
( u  e.  F  <->  u  e.  { <. A , 
( F `  A
) >. } ) )
2018, 19sylibr 205 1  |-  ( F  Fn  { A }  ->  F  =  { <. A ,  ( F `  A ) >. } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   A.wal 1550    = wceq 1653    e. wcel 1726   _Vcvv 2957   {csn 3815   <.cop 3818   dom cdm 4879   Fun wfun 5449    Fn wfn 5450   ` cfv 5455
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 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463
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