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Theorem gruf 9081
Description: A Grothendieck universe contains all functions on its elements. (Contributed by Mario Carneiro, 10-Jun-2013.)
Assertion
Ref Expression
gruf  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  F  e.  U )

Proof of Theorem gruf
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simp3 990 . . . 4  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  F : A --> U )
21feqmptd 5845 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
3 fvex 5801 . . . 4  |-  ( F `
 x )  e. 
_V
43fnasrn 5989 . . 3  |-  ( x  e.  A  |->  ( F `
 x ) )  =  ran  ( x  e.  A  |->  <. x ,  ( F `  x ) >. )
52, 4syl6eq 2508 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  F  =  ran  ( x  e.  A  |->  <. x ,  ( F `  x )
>. ) )
6 simpl1 991 . . . . 5  |-  ( ( ( U  e.  Univ  /\  A  e.  U  /\  F : A --> U )  /\  x  e.  A
)  ->  U  e.  Univ )
7 gruel 9073 . . . . . . 7  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  x  e.  A )  ->  x  e.  U )
873expa 1188 . . . . . 6  |-  ( ( ( U  e.  Univ  /\  A  e.  U )  /\  x  e.  A
)  ->  x  e.  U )
983adantl3 1146 . . . . 5  |-  ( ( ( U  e.  Univ  /\  A  e.  U  /\  F : A --> U )  /\  x  e.  A
)  ->  x  e.  U )
10 ffvelrn 5942 . . . . . 6  |-  ( ( F : A --> U  /\  x  e.  A )  ->  ( F `  x
)  e.  U )
11103ad2antl3 1152 . . . . 5  |-  ( ( ( U  e.  Univ  /\  A  e.  U  /\  F : A --> U )  /\  x  e.  A
)  ->  ( F `  x )  e.  U
)
12 gruop 9075 . . . . 5  |-  ( ( U  e.  Univ  /\  x  e.  U  /\  ( F `  x )  e.  U )  ->  <. x ,  ( F `  x ) >.  e.  U
)
136, 9, 11, 12syl3anc 1219 . . . 4  |-  ( ( ( U  e.  Univ  /\  A  e.  U  /\  F : A --> U )  /\  x  e.  A
)  ->  <. x ,  ( F `  x
) >.  e.  U )
14 eqid 2451 . . . 4  |-  ( x  e.  A  |->  <. x ,  ( F `  x ) >. )  =  ( x  e.  A  |->  <. x ,  ( F `  x )
>. )
1513, 14fmptd 5968 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  (
x  e.  A  |->  <.
x ,  ( F `
 x ) >.
) : A --> U )
16 grurn 9071 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  (
x  e.  A  |->  <.
x ,  ( F `
 x ) >.
) : A --> U )  ->  ran  ( x  e.  A  |->  <. x ,  ( F `  x ) >. )  e.  U )
1715, 16syld3an3 1264 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  ran  ( x  e.  A  |-> 
<. x ,  ( F `
 x ) >.
)  e.  U )
185, 17eqeltrd 2539 1  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  F  e.  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    e. wcel 1758   <.cop 3983    |-> cmpt 4450   ran crn 4941   -->wf 5514   ` cfv 5518   Univcgru 9060
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-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-map 7318  df-gru 9061
This theorem is referenced by: (None)
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