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Theorem bnj145 29071
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 29070 . . . . 5  |-  ( F  Fn  { A }  ->  ( u  e.  F  ->  u  =  <. A , 
( F `  A
) >. ) )
2 df-fn 5274 . . . . . . . 8  |-  ( F  Fn  { A }  <->  ( Fun  F  /\  dom  F  =  { A }
) )
3 bnj145.1 . . . . . . . . . . 11  |-  A  e. 
_V
43snid 3680 . . . . . . . . . 10  |-  A  e. 
{ A }
5 eleq2 2357 . . . . . . . . . 10  |-  ( dom 
F  =  { A }  ->  ( A  e. 
dom  F  <->  A  e.  { A } ) )
64, 5mpbiri 224 . . . . . . . . 9  |-  ( dom 
F  =  { A }  ->  A  e.  dom  F )
76anim2i 552 . . . . . . . 8  |-  ( ( Fun  F  /\  dom  F  =  { A }
)  ->  ( Fun  F  /\  A  e.  dom  F ) )
82, 7sylbi 187 . . . . . . 7  |-  ( F  Fn  { A }  ->  ( Fun  F  /\  A  e.  dom  F ) )
9 funfvop 5653 . . . . . . 7  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  <. A ,  ( F `
 A ) >.  e.  F )
108, 9syl 15 . . . . . 6  |-  ( F  Fn  { A }  -> 
<. A ,  ( F `
 A ) >.  e.  F )
11 eleq1 2356 . . . . . 6  |-  ( u  =  <. A ,  ( F `  A )
>.  ->  ( u  e.  F  <->  <. A ,  ( F `  A )
>.  e.  F ) )
1210, 11syl5ibrcom 213 . . . . 5  |-  ( F  Fn  { A }  ->  ( u  =  <. A ,  ( F `  A ) >.  ->  u  e.  F ) )
131, 12impbid 183 . . . 4  |-  ( F  Fn  { A }  ->  ( u  e.  F  <->  u  =  <. A ,  ( F `  A )
>. ) )
1413alrimiv 1621 . . 3  |-  ( F  Fn  { A }  ->  A. u ( u  e.  F  <->  u  =  <. A ,  ( F `
 A ) >.
) )
15 elsn 3668 . . . . 5  |-  ( u  e.  { <. A , 
( F `  A
) >. }  <->  u  =  <. A ,  ( F `
 A ) >.
)
1615bibi2i 304 . . . 4  |-  ( ( u  e.  F  <->  u  e.  {
<. A ,  ( F `
 A ) >. } )  <->  ( u  e.  F  <->  u  =  <. A ,  ( F `  A ) >. )
)
1716albii 1556 . . 3  |-  ( A. u ( u  e.  F  <->  u  e.  { <. A ,  ( F `  A ) >. } )  <->  A. u ( u  e.  F  <->  u  =  <. A ,  ( F `  A ) >. )
)
1814, 17sylibr 203 . 2  |-  ( F  Fn  { A }  ->  A. u ( u  e.  F  <->  u  e.  {
<. A ,  ( F `
 A ) >. } ) )
19 dfcleq 2290 . 2  |-  ( F  =  { <. A , 
( F `  A
) >. }  <->  A. u
( u  e.  F  <->  u  e.  { <. A , 
( F `  A
) >. } ) )
2018, 19sylibr 203 1  |-  ( F  Fn  { A }  ->  F  =  { <. A ,  ( F `  A ) >. } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530    = wceq 1632    e. wcel 1696   _Vcvv 2801   {csn 3653   <.cop 3656   dom cdm 4705   Fun wfun 5265    Fn wfn 5266   ` cfv 5271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279
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