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Theorem bnj142 29155
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.)
Assertion
Ref Expression
bnj142  |-  ( F  Fn  { A }  ->  ( u  e.  F  ->  u  =  <. A , 
( F `  A
) >. ) )

Proof of Theorem bnj142
StepHypRef Expression
1 fnresdm 5556 . . . 4  |-  ( F  Fn  { A }  ->  ( F  |`  { A } )  =  F )
2 fnfun 5544 . . . . 5  |-  ( F  Fn  { A }  ->  Fun  F )
3 funressn 5921 . . . . 5  |-  ( Fun 
F  ->  ( F  |` 
{ A } ) 
C_  { <. A , 
( F `  A
) >. } )
42, 3syl 16 . . . 4  |-  ( F  Fn  { A }  ->  ( F  |`  { A } )  C_  { <. A ,  ( F `  A ) >. } )
51, 4eqsstr3d 3385 . . 3  |-  ( F  Fn  { A }  ->  F  C_  { <. A , 
( F `  A
) >. } )
65sseld 3349 . 2  |-  ( F  Fn  { A }  ->  ( u  e.  F  ->  u  e.  { <. A ,  ( F `  A ) >. } ) )
7 elsni 3840 . 2  |-  ( u  e.  { <. A , 
( F `  A
) >. }  ->  u  =  <. A ,  ( F `  A )
>. )
86, 7syl6 32 1  |-  ( F  Fn  { A }  ->  ( u  e.  F  ->  u  =  <. A , 
( F `  A
) >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726    C_ wss 3322   {csn 3816   <.cop 3819    |` cres 4882   Fun wfun 5450    Fn wfn 5451   ` cfv 5456
This theorem is referenced by:  bnj145  29156
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464
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