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Theorem bnj142OLD 29348
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.) (Proof modification is discouraged.) Obsolete as of 29-Dec-2018. This is now incorporated into the proof of fnsnb 6089.
Assertion
Ref Expression
bnj142OLD  |-  ( F  Fn  { A }  ->  ( u  e.  F  ->  u  =  <. A , 
( F `  A
) >. ) )

Proof of Theorem bnj142OLD
StepHypRef Expression
1 fnresdm 5694 . . . 4  |-  ( F  Fn  { A }  ->  ( F  |`  { A } )  =  F )
2 fnfun 5682 . . . . 5  |-  ( F  Fn  { A }  ->  Fun  F )
3 funressn 6083 . . . . 5  |-  ( Fun 
F  ->  ( F  |` 
{ A } ) 
C_  { <. A , 
( F `  A
) >. } )
42, 3syl 17 . . . 4  |-  ( F  Fn  { A }  ->  ( F  |`  { A } )  C_  { <. A ,  ( F `  A ) >. } )
51, 4eqsstr3d 3496 . . 3  |-  ( F  Fn  { A }  ->  F  C_  { <. A , 
( F `  A
) >. } )
65sseld 3460 . 2  |-  ( F  Fn  { A }  ->  ( u  e.  F  ->  u  e.  { <. A ,  ( F `  A ) >. } ) )
7 elsni 4018 . 2  |-  ( u  e.  { <. A , 
( F `  A
) >. }  ->  u  =  <. A ,  ( F `  A )
>. )
86, 7syl6 34 1  |-  ( F  Fn  { A }  ->  ( u  e.  F  ->  u  =  <. A , 
( F `  A
) >. ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1867    C_ wss 3433   {csn 3993   <.cop 3999    |` cres 4847   Fun wfun 5586    Fn wfn 5587   ` cfv 5592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600
This theorem is referenced by:  bnj145OLD  29349
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