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Theorem gruima 8424
Description: A Grothendieck's 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 959 . . . 4  |-  ( ( ( U  e.  Univ  /\ 
Fun  F  /\  ( F " A )  C_  U )  /\  A  e.  U )  ->  Fun  F )
2 funrel 5272 . . . 4  |-  ( Fun 
F  ->  Rel  F )
3 resres 4968 . . . . . . 7  |-  ( ( F  |`  dom  F )  |`  A )  =  ( F  |`  ( dom  F  i^i  A ) )
4 resdm 4993 . . . . . . . 8  |-  ( Rel 
F  ->  ( F  |` 
dom  F )  =  F )
54reseq1d 4954 . . . . . . 7  |-  ( Rel 
F  ->  ( ( F  |`  dom  F )  |`  A )  =  ( F  |`  A )
)
63, 5syl5eqr 2329 . . . . . 6  |-  ( Rel 
F  ->  ( F  |`  ( dom  F  i^i  A ) )  =  ( F  |`  A )
)
76rneqd 4906 . . . . 5  |-  ( Rel 
F  ->  ran  ( F  |`  ( dom  F  i^i  A ) )  =  ran  ( F  |`  A ) )
8 df-ima 4702 . . . . 5  |-  ( F
" A )  =  ran  ( F  |`  A )
97, 8syl6reqr 2334 . . . 4  |-  ( Rel 
F  ->  ( F " A )  =  ran  ( F  |`  ( dom 
F  i^i  A )
) )
101, 2, 93syl 18 . . 3  |-  ( ( ( U  e.  Univ  /\ 
Fun  F  /\  ( F " A )  C_  U )  /\  A  e.  U )  ->  ( F " A )  =  ran  ( F  |`  ( dom  F  i^i  A
) ) )
11 simpl1 958 . . . 4  |-  ( ( ( U  e.  Univ  /\ 
Fun  F  /\  ( F " A )  C_  U )  /\  A  e.  U )  ->  U  e.  Univ )
12 simpr 447 . . . . 5  |-  ( ( ( U  e.  Univ  /\ 
Fun  F  /\  ( F " A )  C_  U )  /\  A  e.  U )  ->  A  e.  U )
13 inss2 3390 . . . . . 6  |-  ( dom 
F  i^i  A )  C_  A
1413a1i 10 . . . . 5  |-  ( ( ( U  e.  Univ  /\ 
Fun  F  /\  ( F " A )  C_  U )  /\  A  e.  U )  ->  ( dom  F  i^i  A ) 
C_  A )
15 gruss 8418 . . . . 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 1182 . . . 4  |-  ( ( ( U  e.  Univ  /\ 
Fun  F  /\  ( F " A )  C_  U )  /\  A  e.  U )  ->  ( dom  F  i^i  A )  e.  U )
17 funforn 5458 . . . . . . . 8  |-  ( Fun 
F  <->  F : dom  F -onto-> ran  F )
18 fof 5451 . . . . . . . 8  |-  ( F : dom  F -onto-> ran  F  ->  F : dom  F --> ran  F )
1917, 18sylbi 187 . . . . . . 7  |-  ( Fun 
F  ->  F : dom  F --> ran  F )
20 inss1 3389 . . . . . . 7  |-  ( dom 
F  i^i  A )  C_ 
dom  F
21 fssres 5408 . . . . . . 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 643 . . . . . 6  |-  ( Fun 
F  ->  ( F  |`  ( dom  F  i^i  A ) ) : ( dom  F  i^i  A
) --> ran  F )
23 ffn 5389 . . . . . 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 18 . . . . 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 960 . . . . . 6  |-  ( ( ( U  e.  Univ  /\ 
Fun  F  /\  ( F " A )  C_  U )  /\  A  e.  U )  ->  ( F " A )  C_  U )
2610, 25eqsstr3d 3213 . . . . 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 5259 . . . . 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 645 . . . 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 8423 . . . 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 1182 . . 3  |-  ( ( ( U  e.  Univ  /\ 
Fun  F  /\  ( F " A )  C_  U )  /\  A  e.  U )  ->  ran  ( F  |`  ( dom 
F  i^i  A )
)  e.  U )
3110, 30eqeltrd 2357 . 2  |-  ( ( ( U  e.  Univ  /\ 
Fun  F  /\  ( F " A )  C_  U )  /\  A  e.  U )  ->  ( F " A )  e.  U )
3231ex 423 1  |-  ( ( U  e.  Univ  /\  Fun  F  /\  ( F " A )  C_  U
)  ->  ( A  e.  U  ->  ( F
" A )  e.  U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692   Rel wrel 4694   Fun wfun 5249    Fn wfn 5250   -->wf 5251   -onto->wfo 5253   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-rab 2552  df-v 2790  df-sbc 2992  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-br 4024  df-opab 4078  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-fo 5261  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-gru 8413
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