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Theorem gruima 8440
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 5288 . . . 4  |-  ( Fun 
F  ->  Rel  F )
3 resres 4984 . . . . . . 7  |-  ( ( F  |`  dom  F )  |`  A )  =  ( F  |`  ( dom  F  i^i  A ) )
4 resdm 5009 . . . . . . . 8  |-  ( Rel 
F  ->  ( F  |` 
dom  F )  =  F )
54reseq1d 4970 . . . . . . 7  |-  ( Rel 
F  ->  ( ( F  |`  dom  F )  |`  A )  =  ( F  |`  A )
)
63, 5syl5eqr 2342 . . . . . 6  |-  ( Rel 
F  ->  ( F  |`  ( dom  F  i^i  A ) )  =  ( F  |`  A )
)
76rneqd 4922 . . . . 5  |-  ( Rel 
F  ->  ran  ( F  |`  ( dom  F  i^i  A ) )  =  ran  ( F  |`  A ) )
8 df-ima 4718 . . . . 5  |-  ( F
" A )  =  ran  ( F  |`  A )
97, 8syl6reqr 2347 . . . 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 3403 . . . . . 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 8434 . . . . 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 5474 . . . . . . . 8  |-  ( Fun 
F  <->  F : dom  F -onto-> ran  F )
18 fof 5467 . . . . . . . 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 3402 . . . . . . 7  |-  ( dom 
F  i^i  A )  C_ 
dom  F
21 fssres 5424 . . . . . . 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 5405 . . . . . 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 3226 . . . . 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 5275 . . . . 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 8439 . . . 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 2370 . 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 1632    e. wcel 1696    i^i cin 3164    C_ wss 3165   dom cdm 4705   ran crn 4706    |` cres 4707   "cima 4708   Rel wrel 4710   Fun wfun 5265    Fn wfn 5266   -->wf 5267   -onto->wfo 5269   Univcgru 8428
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-gru 8429
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