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Theorem gruima 9180
Description: A Grothendieck universe contains image sets drawn from its members. (Contributed by Mario Carneiro, 9-Jun-2013.)
Assertion
Ref Expression
gruima  |-  ( ( U  e.  Univ  /\  Fun  F  /\  ( F " A )  C_  U
)  ->  ( A  e.  U  ->  ( F
" A )  e.  U ) )

Proof of Theorem gruima
StepHypRef Expression
1 simpl2 1000 . . . 4  |-  ( ( ( U  e.  Univ  /\ 
Fun  F  /\  ( F " A )  C_  U )  /\  A  e.  U )  ->  Fun  F )
2 funrel 5605 . . . 4  |-  ( Fun 
F  ->  Rel  F )
3 resres 5286 . . . . . . 7  |-  ( ( F  |`  dom  F )  |`  A )  =  ( F  |`  ( dom  F  i^i  A ) )
4 resdm 5315 . . . . . . . 8  |-  ( Rel 
F  ->  ( F  |` 
dom  F )  =  F )
54reseq1d 5272 . . . . . . 7  |-  ( Rel 
F  ->  ( ( F  |`  dom  F )  |`  A )  =  ( F  |`  A )
)
63, 5syl5eqr 2522 . . . . . 6  |-  ( Rel 
F  ->  ( F  |`  ( dom  F  i^i  A ) )  =  ( F  |`  A )
)
76rneqd 5230 . . . . 5  |-  ( Rel 
F  ->  ran  ( F  |`  ( dom  F  i^i  A ) )  =  ran  ( F  |`  A ) )
8 df-ima 5012 . . . . 5  |-  ( F
" A )  =  ran  ( F  |`  A )
97, 8syl6reqr 2527 . . . 4  |-  ( Rel 
F  ->  ( F " A )  =  ran  ( F  |`  ( dom 
F  i^i  A )
) )
101, 2, 93syl 20 . . 3  |-  ( ( ( U  e.  Univ  /\ 
Fun  F  /\  ( F " A )  C_  U )  /\  A  e.  U )  ->  ( F " A )  =  ran  ( F  |`  ( dom  F  i^i  A
) ) )
11 simpl1 999 . . . 4  |-  ( ( ( U  e.  Univ  /\ 
Fun  F  /\  ( F " A )  C_  U )  /\  A  e.  U )  ->  U  e.  Univ )
12 simpr 461 . . . . 5  |-  ( ( ( U  e.  Univ  /\ 
Fun  F  /\  ( F " A )  C_  U )  /\  A  e.  U )  ->  A  e.  U )
13 inss2 3719 . . . . . 6  |-  ( dom 
F  i^i  A )  C_  A
1413a1i 11 . . . . 5  |-  ( ( ( U  e.  Univ  /\ 
Fun  F  /\  ( F " A )  C_  U )  /\  A  e.  U )  ->  ( dom  F  i^i  A ) 
C_  A )
15 gruss 9174 . . . . 5  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  ( dom  F  i^i  A ) 
C_  A )  -> 
( dom  F  i^i  A )  e.  U )
1611, 12, 14, 15syl3anc 1228 . . . 4  |-  ( ( ( U  e.  Univ  /\ 
Fun  F  /\  ( F " A )  C_  U )  /\  A  e.  U )  ->  ( dom  F  i^i  A )  e.  U )
17 funforn 5802 . . . . . . . 8  |-  ( Fun 
F  <->  F : dom  F -onto-> ran  F )
18 fof 5795 . . . . . . . 8  |-  ( F : dom  F -onto-> ran  F  ->  F : dom  F --> ran  F )
1917, 18sylbi 195 . . . . . . 7  |-  ( Fun 
F  ->  F : dom  F --> ran  F )
20 inss1 3718 . . . . . . 7  |-  ( dom 
F  i^i  A )  C_ 
dom  F
21 fssres 5751 . . . . . . 7  |-  ( ( F : dom  F --> ran  F  /\  ( dom 
F  i^i  A )  C_ 
dom  F )  -> 
( F  |`  ( dom  F  i^i  A ) ) : ( dom 
F  i^i  A ) --> ran  F )
2219, 20, 21sylancl 662 . . . . . 6  |-  ( Fun 
F  ->  ( F  |`  ( dom  F  i^i  A ) ) : ( dom  F  i^i  A
) --> ran  F )
23 ffn 5731 . . . . . 6  |-  ( ( F  |`  ( dom  F  i^i  A ) ) : ( dom  F  i^i  A ) --> ran  F  ->  ( F  |`  ( dom  F  i^i  A ) )  Fn  ( dom 
F  i^i  A )
)
241, 22, 233syl 20 . . . . 5  |-  ( ( ( U  e.  Univ  /\ 
Fun  F  /\  ( F " A )  C_  U )  /\  A  e.  U )  ->  ( F  |`  ( dom  F  i^i  A ) )  Fn  ( dom  F  i^i  A ) )
25 simpl3 1001 . . . . . 6  |-  ( ( ( U  e.  Univ  /\ 
Fun  F  /\  ( F " A )  C_  U )  /\  A  e.  U )  ->  ( F " A )  C_  U )
2610, 25eqsstr3d 3539 . . . . 5  |-  ( ( ( U  e.  Univ  /\ 
Fun  F  /\  ( F " A )  C_  U )  /\  A  e.  U )  ->  ran  ( F  |`  ( dom 
F  i^i  A )
)  C_  U )
27 df-f 5592 . . . . 5  |-  ( ( F  |`  ( dom  F  i^i  A ) ) : ( dom  F  i^i  A ) --> U  <->  ( ( F  |`  ( dom  F  i^i  A ) )  Fn  ( dom  F  i^i  A )  /\  ran  ( F  |`  ( dom  F  i^i  A ) )  C_  U ) )
2824, 26, 27sylanbrc 664 . . . 4  |-  ( ( ( U  e.  Univ  /\ 
Fun  F  /\  ( F " A )  C_  U )  /\  A  e.  U )  ->  ( F  |`  ( dom  F  i^i  A ) ) : ( dom  F  i^i  A ) --> U )
29 grurn 9179 . . . 4  |-  ( ( U  e.  Univ  /\  ( dom  F  i^i  A )  e.  U  /\  ( F  |`  ( dom  F  i^i  A ) ) : ( dom  F  i^i  A ) --> U )  ->  ran  ( F  |`  ( dom  F  i^i  A ) )  e.  U )
3011, 16, 28, 29syl3anc 1228 . . 3  |-  ( ( ( U  e.  Univ  /\ 
Fun  F  /\  ( F " A )  C_  U )  /\  A  e.  U )  ->  ran  ( F  |`  ( dom 
F  i^i  A )
)  e.  U )
3110, 30eqeltrd 2555 . 2  |-  ( ( ( U  e.  Univ  /\ 
Fun  F  /\  ( F " A )  C_  U )  /\  A  e.  U )  ->  ( F " A )  e.  U )
3231ex 434 1  |-  ( ( U  e.  Univ  /\  Fun  F  /\  ( F " A )  C_  U
)  ->  ( A  e.  U  ->  ( F
" A )  e.  U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    i^i cin 3475    C_ wss 3476   dom cdm 4999   ran crn 5000    |` cres 5001   "cima 5002   Rel wrel 5004   Fun wfun 5582    Fn wfn 5583   -->wf 5584   -onto->wfo 5586   Univcgru 9168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fo 5594  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-map 7422  df-gru 9169
This theorem is referenced by: (None)
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