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Theorem bnj145 29031
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 29030 . . . . 5  |-  ( F  Fn  { A }  ->  ( u  e.  F  ->  u  =  <. A , 
( F `  A
) >. ) )
2 df-fn 5449 . . . . . . . 8  |-  ( F  Fn  { A }  <->  ( Fun  F  /\  dom  F  =  { A }
) )
3 bnj145.1 . . . . . . . . . . 11  |-  A  e. 
_V
43snid 3833 . . . . . . . . . 10  |-  A  e. 
{ A }
5 eleq2 2496 . . . . . . . . . 10  |-  ( dom 
F  =  { A }  ->  ( A  e. 
dom  F  <->  A  e.  { A } ) )
64, 5mpbiri 225 . . . . . . . . 9  |-  ( dom 
F  =  { A }  ->  A  e.  dom  F )
76anim2i 553 . . . . . . . 8  |-  ( ( Fun  F  /\  dom  F  =  { A }
)  ->  ( Fun  F  /\  A  e.  dom  F ) )
82, 7sylbi 188 . . . . . . 7  |-  ( F  Fn  { A }  ->  ( Fun  F  /\  A  e.  dom  F ) )
9 funfvop 5834 . . . . . . 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 2495 . . . . . 6  |-  ( u  =  <. A ,  ( F `  A )
>.  ->  ( u  e.  F  <->  <. A ,  ( F `  A )
>.  e.  F ) )
1210, 11syl5ibrcom 214 . . . . 5  |-  ( F  Fn  { A }  ->  ( u  =  <. A ,  ( F `  A ) >.  ->  u  e.  F ) )
131, 12impbid 184 . . . 4  |-  ( F  Fn  { A }  ->  ( u  e.  F  <->  u  =  <. A ,  ( F `  A )
>. ) )
1413alrimiv 1641 . . 3  |-  ( F  Fn  { A }  ->  A. u ( u  e.  F  <->  u  =  <. A ,  ( F `
 A ) >.
) )
15 elsn 3821 . . . . 5  |-  ( u  e.  { <. A , 
( F `  A
) >. }  <->  u  =  <. A ,  ( F `
 A ) >.
)
1615bibi2i 305 . . . 4  |-  ( ( u  e.  F  <->  u  e.  {
<. A ,  ( F `
 A ) >. } )  <->  ( u  e.  F  <->  u  =  <. A ,  ( F `  A ) >. )
)
1716albii 1575 . . 3  |-  ( A. u ( u  e.  F  <->  u  e.  { <. A ,  ( F `  A ) >. } )  <->  A. u ( u  e.  F  <->  u  =  <. A ,  ( F `  A ) >. )
)
1814, 17sylibr 204 . 2  |-  ( F  Fn  { A }  ->  A. u ( u  e.  F  <->  u  e.  {
<. A ,  ( F `
 A ) >. } ) )
19 dfcleq 2429 . 2  |-  ( F  =  { <. A , 
( F `  A
) >. }  <->  A. u
( u  e.  F  <->  u  e.  { <. A , 
( F `  A
) >. } ) )
2018, 19sylibr 204 1  |-  ( F  Fn  { A }  ->  F  =  { <. A ,  ( F `  A ) >. } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549    = wceq 1652    e. wcel 1725   _Vcvv 2948   {csn 3806   <.cop 3809   dom cdm 4870   Fun wfun 5440    Fn wfn 5441   ` cfv 5446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454
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