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Theorem bnj145OLD 32070
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.) Obsolete as of 29-Dec-2018. This is now incorporated into the proof of fnsnb 6010.
Hypotheses
Ref Expression
bnj145OLD.1  |-  A  e. 
_V
bnj145OLD.2  |-  ( F `
 A )  e. 
_V
Assertion
Ref Expression
bnj145OLD  |-  ( F  Fn  { A }  ->  F  =  { <. A ,  ( F `  A ) >. } )

Proof of Theorem bnj145OLD
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 bnj142OLD 32069 . . . . 5  |-  ( F  Fn  { A }  ->  ( u  e.  F  ->  u  =  <. A , 
( F `  A
) >. ) )
2 df-fn 5532 . . . . . . . 8  |-  ( F  Fn  { A }  <->  ( Fun  F  /\  dom  F  =  { A }
) )
3 bnj145OLD.1 . . . . . . . . . . 11  |-  A  e. 
_V
43snid 4016 . . . . . . . . . 10  |-  A  e. 
{ A }
5 eleq2 2527 . . . . . . . . . 10  |-  ( dom 
F  =  { A }  ->  ( A  e. 
dom  F  <->  A  e.  { A } ) )
64, 5mpbiri 233 . . . . . . . . 9  |-  ( dom 
F  =  { A }  ->  A  e.  dom  F )
76anim2i 569 . . . . . . . 8  |-  ( ( Fun  F  /\  dom  F  =  { A }
)  ->  ( Fun  F  /\  A  e.  dom  F ) )
82, 7sylbi 195 . . . . . . 7  |-  ( F  Fn  { A }  ->  ( Fun  F  /\  A  e.  dom  F ) )
9 funfvop 5927 . . . . . . 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 2526 . . . . . 6  |-  ( u  =  <. A ,  ( F `  A )
>.  ->  ( u  e.  F  <->  <. A ,  ( F `  A )
>.  e.  F ) )
1210, 11syl5ibrcom 222 . . . . 5  |-  ( F  Fn  { A }  ->  ( u  =  <. A ,  ( F `  A ) >.  ->  u  e.  F ) )
131, 12impbid 191 . . . 4  |-  ( F  Fn  { A }  ->  ( u  e.  F  <->  u  =  <. A ,  ( F `  A )
>. ) )
1413alrimiv 1686 . . 3  |-  ( F  Fn  { A }  ->  A. u ( u  e.  F  <->  u  =  <. A ,  ( F `
 A ) >.
) )
15 elsn 4002 . . . . 5  |-  ( u  e.  { <. A , 
( F `  A
) >. }  <->  u  =  <. A ,  ( F `
 A ) >.
)
1615bibi2i 313 . . . 4  |-  ( ( u  e.  F  <->  u  e.  {
<. A ,  ( F `
 A ) >. } )  <->  ( u  e.  F  <->  u  =  <. A ,  ( F `  A ) >. )
)
1716albii 1611 . . 3  |-  ( A. u ( u  e.  F  <->  u  e.  { <. A ,  ( F `  A ) >. } )  <->  A. u ( u  e.  F  <->  u  =  <. A ,  ( F `  A ) >. )
)
1814, 17sylibr 212 . 2  |-  ( F  Fn  { A }  ->  A. u ( u  e.  F  <->  u  e.  {
<. A ,  ( F `
 A ) >. } ) )
19 dfcleq 2447 . 2  |-  ( F  =  { <. A , 
( F `  A
) >. }  <->  A. u
( u  e.  F  <->  u  e.  { <. A , 
( F `  A
) >. } ) )
2018, 19sylibr 212 1  |-  ( F  Fn  { A }  ->  F  =  { <. A ,  ( F `  A ) >. } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1368    = wceq 1370    e. wcel 1758   _Vcvv 3078   {csn 3988   <.cop 3994   dom cdm 4951   Fun wfun 5523    Fn wfn 5524   ` cfv 5529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537
This theorem is referenced by: (None)
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