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Theorem bnj142OLD 31740
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 5917.
Assertion
Ref Expression
bnj142OLD  |-  ( F  Fn  { A }  ->  ( u  e.  F  ->  u  =  <. A , 
( F `  A
) >. ) )

Proof of Theorem bnj142OLD
StepHypRef Expression
1 fnresdm 5539 . . . 4  |-  ( F  Fn  { A }  ->  ( F  |`  { A } )  =  F )
2 fnfun 5527 . . . . 5  |-  ( F  Fn  { A }  ->  Fun  F )
3 funressn 5914 . . . . 5  |-  ( Fun 
F  ->  ( F  |` 
{ A } ) 
C_  { <. A , 
( F `  A
) >. } )
42, 3syl 16 . . . 4  |-  ( F  Fn  { A }  ->  ( F  |`  { A } )  C_  { <. A ,  ( F `  A ) >. } )
51, 4eqsstr3d 3410 . . 3  |-  ( F  Fn  { A }  ->  F  C_  { <. A , 
( F `  A
) >. } )
65sseld 3374 . 2  |-  ( F  Fn  { A }  ->  ( u  e.  F  ->  u  e.  { <. A ,  ( F `  A ) >. } ) )
7 elsni 3921 . 2  |-  ( u  e.  { <. A , 
( F `  A
) >. }  ->  u  =  <. A ,  ( F `  A )
>. )
86, 7syl6 33 1  |-  ( F  Fn  { A }  ->  ( u  e.  F  ->  u  =  <. A , 
( F `  A
) >. ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756    C_ wss 3347   {csn 3896   <.cop 3902    |` cres 4861   Fun wfun 5431    Fn wfn 5432   ` cfv 5437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pr 4550
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-reu 2741  df-rab 2743  df-v 2993  df-sbc 3206  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-br 4312  df-opab 4370  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445
This theorem is referenced by:  bnj145OLD  31741
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