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Theorem gruf 8580
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 958 . . . 4  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  F : A --> U )
21feqmptd 5682 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
3 fvex 5646 . . . 4  |-  ( F `
 x )  e. 
_V
43fnasrn 5813 . . 3  |-  ( x  e.  A  |->  ( F `
 x ) )  =  ran  ( x  e.  A  |->  <. x ,  ( F `  x ) >. )
52, 4syl6eq 2414 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  F  =  ran  ( x  e.  A  |->  <. x ,  ( F `  x )
>. ) )
6 simpl1 959 . . . . 5  |-  ( ( ( U  e.  Univ  /\  A  e.  U  /\  F : A --> U )  /\  x  e.  A
)  ->  U  e.  Univ )
7 gruel 8572 . . . . . . 7  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  x  e.  A )  ->  x  e.  U )
873expa 1152 . . . . . 6  |-  ( ( ( U  e.  Univ  /\  A  e.  U )  /\  x  e.  A
)  ->  x  e.  U )
983adantl3 1114 . . . . 5  |-  ( ( ( U  e.  Univ  /\  A  e.  U  /\  F : A --> U )  /\  x  e.  A
)  ->  x  e.  U )
10 ffvelrn 5770 . . . . . 6  |-  ( ( F : A --> U  /\  x  e.  A )  ->  ( F `  x
)  e.  U )
11103ad2antl3 1120 . . . . 5  |-  ( ( ( U  e.  Univ  /\  A  e.  U  /\  F : A --> U )  /\  x  e.  A
)  ->  ( F `  x )  e.  U
)
12 gruop 8574 . . . . 5  |-  ( ( U  e.  Univ  /\  x  e.  U  /\  ( F `  x )  e.  U )  ->  <. x ,  ( F `  x ) >.  e.  U
)
136, 9, 11, 12syl3anc 1183 . . . 4  |-  ( ( ( U  e.  Univ  /\  A  e.  U  /\  F : A --> U )  /\  x  e.  A
)  ->  <. x ,  ( F `  x
) >.  e.  U )
14 eqid 2366 . . . 4  |-  ( x  e.  A  |->  <. x ,  ( F `  x ) >. )  =  ( x  e.  A  |->  <. x ,  ( F `  x )
>. )
1513, 14fmptd 5795 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  (
x  e.  A  |->  <.
x ,  ( F `
 x ) >.
) : A --> U )
16 grurn 8570 . . 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 1228 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  ran  ( x  e.  A  |-> 
<. x ,  ( F `
 x ) >.
)  e.  U )
185, 17eqeltrd 2440 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 935    e. wcel 1715   <.cop 3732    e. cmpt 4179   ran crn 4793   -->wf 5354   ` cfv 5358   Univcgru 8559
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-map 6917  df-gru 8560
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