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Theorem gruf 9178
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 996 . . . 4  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  F : A --> U )
21feqmptd 5901 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
3 fvex 5858 . . . 4  |-  ( F `
 x )  e. 
_V
43fnasrn 6053 . . 3  |-  ( x  e.  A  |->  ( F `
 x ) )  =  ran  ( x  e.  A  |->  <. x ,  ( F `  x ) >. )
52, 4syl6eq 2511 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  F  =  ran  ( x  e.  A  |->  <. x ,  ( F `  x )
>. ) )
6 simpl1 997 . . . . 5  |-  ( ( ( U  e.  Univ  /\  A  e.  U  /\  F : A --> U )  /\  x  e.  A
)  ->  U  e.  Univ )
7 gruel 9170 . . . . . . 7  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  x  e.  A )  ->  x  e.  U )
873expa 1194 . . . . . 6  |-  ( ( ( U  e.  Univ  /\  A  e.  U )  /\  x  e.  A
)  ->  x  e.  U )
983adantl3 1152 . . . . 5  |-  ( ( ( U  e.  Univ  /\  A  e.  U  /\  F : A --> U )  /\  x  e.  A
)  ->  x  e.  U )
10 ffvelrn 6005 . . . . . 6  |-  ( ( F : A --> U  /\  x  e.  A )  ->  ( F `  x
)  e.  U )
11103ad2antl3 1158 . . . . 5  |-  ( ( ( U  e.  Univ  /\  A  e.  U  /\  F : A --> U )  /\  x  e.  A
)  ->  ( F `  x )  e.  U
)
12 gruop 9172 . . . . 5  |-  ( ( U  e.  Univ  /\  x  e.  U  /\  ( F `  x )  e.  U )  ->  <. x ,  ( F `  x ) >.  e.  U
)
136, 9, 11, 12syl3anc 1226 . . . 4  |-  ( ( ( U  e.  Univ  /\  A  e.  U  /\  F : A --> U )  /\  x  e.  A
)  ->  <. x ,  ( F `  x
) >.  e.  U )
14 eqid 2454 . . . 4  |-  ( x  e.  A  |->  <. x ,  ( F `  x ) >. )  =  ( x  e.  A  |->  <. x ,  ( F `  x )
>. )
1513, 14fmptd 6031 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  (
x  e.  A  |->  <.
x ,  ( F `
 x ) >.
) : A --> U )
16 grurn 9168 . . 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 1271 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  ran  ( x  e.  A  |-> 
<. x ,  ( F `
 x ) >.
)  e.  U )
185, 17eqeltrd 2542 1  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A
--> U )  ->  F  e.  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    e. wcel 1823   <.cop 4022    |-> cmpt 4497   ran crn 4989   -->wf 5566   ` cfv 5570   Univcgru 9157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-map 7414  df-gru 9158
This theorem is referenced by: (None)
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