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Theorem gruima 9197
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 5611 . . . 4  |-  ( Fun 
F  ->  Rel  F )
3 resres 5296 . . . . . . 7  |-  ( ( F  |`  dom  F )  |`  A )  =  ( F  |`  ( dom  F  i^i  A ) )
4 resdm 5325 . . . . . . . 8  |-  ( Rel 
F  ->  ( F  |` 
dom  F )  =  F )
54reseq1d 5282 . . . . . . 7  |-  ( Rel 
F  ->  ( ( F  |`  dom  F )  |`  A )  =  ( F  |`  A )
)
63, 5syl5eqr 2512 . . . . . 6  |-  ( Rel 
F  ->  ( F  |`  ( dom  F  i^i  A ) )  =  ( F  |`  A )
)
76rneqd 5240 . . . . 5  |-  ( Rel 
F  ->  ran  ( F  |`  ( dom  F  i^i  A ) )  =  ran  ( F  |`  A ) )
8 df-ima 5021 . . . . 5  |-  ( F
" A )  =  ran  ( F  |`  A )
97, 8syl6reqr 2517 . . . 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 3715 . . . . . 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 9191 . . . . 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 5808 . . . . . . . 8  |-  ( Fun 
F  <->  F : dom  F -onto-> ran  F )
18 fof 5801 . . . . . . . 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 3714 . . . . . . 7  |-  ( dom 
F  i^i  A )  C_ 
dom  F
21 fssres 5757 . . . . . . 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 5737 . . . . . 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 3534 . . . . 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 5598 . . . . 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 9196 . . . 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 2545 . 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 1395    e. wcel 1819    i^i cin 3470    C_ wss 3471   dom cdm 5008   ran crn 5009    |` cres 5010   "cima 5011   Rel wrel 5013   Fun wfun 5588    Fn wfn 5589   -->wf 5590   -onto->wfo 5592   Univcgru 9185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-tr 4551  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fo 5600  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-map 7440  df-gru 9186
This theorem is referenced by: (None)
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