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Theorem gruf 8433
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 957 . . . 4  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  F : A --> U )
21feqmptd 5575 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
3 fvex 5539 . . . 4  |-  ( F `
 x )  e. 
_V
43fnasrn 5702 . . 3  |-  ( x  e.  A  |->  ( F `
 x ) )  =  ran  ( x  e.  A  |->  <. x ,  ( F `  x ) >. )
52, 4syl6eq 2331 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  F  =  ran  ( x  e.  A  |->  <. x ,  ( F `  x )
>. ) )
6 simpl1 958 . . . . 5  |-  ( ( ( U  e.  Univ  /\  A  e.  U  /\  F : A --> U )  /\  x  e.  A
)  ->  U  e.  Univ )
7 gruel 8425 . . . . . . 7  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  x  e.  A )  ->  x  e.  U )
873expa 1151 . . . . . 6  |-  ( ( ( U  e.  Univ  /\  A  e.  U )  /\  x  e.  A
)  ->  x  e.  U )
983adantl3 1113 . . . . 5  |-  ( ( ( U  e.  Univ  /\  A  e.  U  /\  F : A --> U )  /\  x  e.  A
)  ->  x  e.  U )
10 ffvelrn 5663 . . . . . 6  |-  ( ( F : A --> U  /\  x  e.  A )  ->  ( F `  x
)  e.  U )
11103ad2antl3 1119 . . . . 5  |-  ( ( ( U  e.  Univ  /\  A  e.  U  /\  F : A --> U )  /\  x  e.  A
)  ->  ( F `  x )  e.  U
)
12 gruop 8427 . . . . 5  |-  ( ( U  e.  Univ  /\  x  e.  U  /\  ( F `  x )  e.  U )  ->  <. x ,  ( F `  x ) >.  e.  U
)
136, 9, 11, 12syl3anc 1182 . . . 4  |-  ( ( ( U  e.  Univ  /\  A  e.  U  /\  F : A --> U )  /\  x  e.  A
)  ->  <. x ,  ( F `  x
) >.  e.  U )
14 eqid 2283 . . . 4  |-  ( x  e.  A  |->  <. x ,  ( F `  x ) >. )  =  ( x  e.  A  |->  <. x ,  ( F `  x )
>. )
1513, 14fmptd 5684 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  (
x  e.  A  |->  <.
x ,  ( F `
 x ) >.
) : A --> U )
16 grurn 8423 . . 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 1227 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  ran  ( x  e.  A  |-> 
<. x ,  ( F `
 x ) >.
)  e.  U )
185, 17eqeltrd 2357 1  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  F  e.  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    e. wcel 1684   <.cop 3643    e. cmpt 4077   ran crn 4690   -->wf 5251   ` cfv 5255   Univcgru 8412
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-13 1686  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-pow 4188  ax-pr 4214  ax-un 4512
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-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  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  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-gru 8413
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