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Theorem bnj142 28754
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 5353 . . . 4  |-  ( F  Fn  { A }  ->  ( F  |`  { A } )  =  F )
2 fnfun 5341 . . . . 5  |-  ( F  Fn  { A }  ->  Fun  F )
3 funressn 5706 . . . . 5  |-  ( Fun 
F  ->  ( F  |` 
{ A } ) 
C_  { <. A , 
( F `  A
) >. } )
42, 3syl 15 . . . 4  |-  ( F  Fn  { A }  ->  ( F  |`  { A } )  C_  { <. A ,  ( F `  A ) >. } )
51, 4eqsstr3d 3213 . . 3  |-  ( F  Fn  { A }  ->  F  C_  { <. A , 
( F `  A
) >. } )
65sseld 3179 . 2  |-  ( F  Fn  { A }  ->  ( u  e.  F  ->  u  e.  { <. A ,  ( F `  A ) >. } ) )
7 elsni 3664 . 2  |-  ( u  e.  { <. A , 
( F `  A
) >. }  ->  u  =  <. A ,  ( F `  A )
>. )
86, 7syl6 29 1  |-  ( F  Fn  { A }  ->  ( u  e.  F  ->  u  =  <. A , 
( F `  A
) >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    C_ wss 3152   {csn 3640   <.cop 3643    |` cres 4691   Fun wfun 5249    Fn wfn 5250   ` cfv 5255
This theorem is referenced by:  bnj145  28755
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263
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