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Theorem gruf 8717
Description: A Grothendieck's 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 960 . . . 4  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  F : A --> U )
21feqmptd 5808 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
3 fvex 5771 . . . 4  |-  ( F `
 x )  e. 
_V
43fnasrn 5941 . . 3  |-  ( x  e.  A  |->  ( F `
 x ) )  =  ran  ( x  e.  A  |->  <. x ,  ( F `  x ) >. )
52, 4syl6eq 2490 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  F  =  ran  ( x  e.  A  |->  <. x ,  ( F `  x )
>. ) )
6 simpl1 961 . . . . 5  |-  ( ( ( U  e.  Univ  /\  A  e.  U  /\  F : A --> U )  /\  x  e.  A
)  ->  U  e.  Univ )
7 gruel 8709 . . . . . . 7  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  x  e.  A )  ->  x  e.  U )
873expa 1154 . . . . . 6  |-  ( ( ( U  e.  Univ  /\  A  e.  U )  /\  x  e.  A
)  ->  x  e.  U )
983adantl3 1116 . . . . 5  |-  ( ( ( U  e.  Univ  /\  A  e.  U  /\  F : A --> U )  /\  x  e.  A
)  ->  x  e.  U )
10 ffvelrn 5897 . . . . . 6  |-  ( ( F : A --> U  /\  x  e.  A )  ->  ( F `  x
)  e.  U )
11103ad2antl3 1122 . . . . 5  |-  ( ( ( U  e.  Univ  /\  A  e.  U  /\  F : A --> U )  /\  x  e.  A
)  ->  ( F `  x )  e.  U
)
12 gruop 8711 . . . . 5  |-  ( ( U  e.  Univ  /\  x  e.  U  /\  ( F `  x )  e.  U )  ->  <. x ,  ( F `  x ) >.  e.  U
)
136, 9, 11, 12syl3anc 1185 . . . 4  |-  ( ( ( U  e.  Univ  /\  A  e.  U  /\  F : A --> U )  /\  x  e.  A
)  ->  <. x ,  ( F `  x
) >.  e.  U )
14 eqid 2442 . . . 4  |-  ( x  e.  A  |->  <. x ,  ( F `  x ) >. )  =  ( x  e.  A  |->  <. x ,  ( F `  x )
>. )
1513, 14fmptd 5922 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  (
x  e.  A  |->  <.
x ,  ( F `
 x ) >.
) : A --> U )
16 grurn 8707 . . 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 1230 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  ran  ( x  e.  A  |-> 
<. x ,  ( F `
 x ) >.
)  e.  U )
185, 17eqeltrd 2516 1  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  F  e.  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    e. wcel 1727   <.cop 3841    e. cmpt 4291   ran crn 4908   -->wf 5479   ` cfv 5483   Univcgru 8696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
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 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-map 7049  df-gru 8697
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