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Theorem gruima 8669
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 961 . . . 4  |-  ( ( ( U  e.  Univ  /\ 
Fun  F  /\  ( F " A )  C_  U )  /\  A  e.  U )  ->  Fun  F )
2 funrel 5463 . . . 4  |-  ( Fun 
F  ->  Rel  F )
3 resres 5151 . . . . . . 7  |-  ( ( F  |`  dom  F )  |`  A )  =  ( F  |`  ( dom  F  i^i  A ) )
4 resdm 5176 . . . . . . . 8  |-  ( Rel 
F  ->  ( F  |` 
dom  F )  =  F )
54reseq1d 5137 . . . . . . 7  |-  ( Rel 
F  ->  ( ( F  |`  dom  F )  |`  A )  =  ( F  |`  A )
)
63, 5syl5eqr 2481 . . . . . 6  |-  ( Rel 
F  ->  ( F  |`  ( dom  F  i^i  A ) )  =  ( F  |`  A )
)
76rneqd 5089 . . . . 5  |-  ( Rel 
F  ->  ran  ( F  |`  ( dom  F  i^i  A ) )  =  ran  ( F  |`  A ) )
8 df-ima 4883 . . . . 5  |-  ( F
" A )  =  ran  ( F  |`  A )
97, 8syl6reqr 2486 . . . 4  |-  ( Rel 
F  ->  ( F " A )  =  ran  ( F  |`  ( dom 
F  i^i  A )
) )
101, 2, 93syl 19 . . 3  |-  ( ( ( U  e.  Univ  /\ 
Fun  F  /\  ( F " A )  C_  U )  /\  A  e.  U )  ->  ( F " A )  =  ran  ( F  |`  ( dom  F  i^i  A
) ) )
11 simpl1 960 . . . 4  |-  ( ( ( U  e.  Univ  /\ 
Fun  F  /\  ( F " A )  C_  U )  /\  A  e.  U )  ->  U  e.  Univ )
12 simpr 448 . . . . 5  |-  ( ( ( U  e.  Univ  /\ 
Fun  F  /\  ( F " A )  C_  U )  /\  A  e.  U )  ->  A  e.  U )
13 inss2 3554 . . . . . 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 8663 . . . . 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 1184 . . . 4  |-  ( ( ( U  e.  Univ  /\ 
Fun  F  /\  ( F " A )  C_  U )  /\  A  e.  U )  ->  ( dom  F  i^i  A )  e.  U )
17 funforn 5652 . . . . . . . 8  |-  ( Fun 
F  <->  F : dom  F -onto-> ran  F )
18 fof 5645 . . . . . . . 8  |-  ( F : dom  F -onto-> ran  F  ->  F : dom  F --> ran  F )
1917, 18sylbi 188 . . . . . . 7  |-  ( Fun 
F  ->  F : dom  F --> ran  F )
20 inss1 3553 . . . . . . 7  |-  ( dom 
F  i^i  A )  C_ 
dom  F
21 fssres 5602 . . . . . . 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 644 . . . . . 6  |-  ( Fun 
F  ->  ( F  |`  ( dom  F  i^i  A ) ) : ( dom  F  i^i  A
) --> ran  F )
23 ffn 5583 . . . . . 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 19 . . . . 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 962 . . . . . 6  |-  ( ( ( U  e.  Univ  /\ 
Fun  F  /\  ( F " A )  C_  U )  /\  A  e.  U )  ->  ( F " A )  C_  U )
2610, 25eqsstr3d 3375 . . . . 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 5450 . . . . 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 646 . . . 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 8668 . . . 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 1184 . . 3  |-  ( ( ( U  e.  Univ  /\ 
Fun  F  /\  ( F " A )  C_  U )  /\  A  e.  U )  ->  ran  ( F  |`  ( dom 
F  i^i  A )
)  e.  U )
3110, 30eqeltrd 2509 . 2  |-  ( ( ( U  e.  Univ  /\ 
Fun  F  /\  ( F " A )  C_  U )  /\  A  e.  U )  ->  ( F " A )  e.  U )
3231ex 424 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725    i^i cin 3311    C_ wss 3312   dom cdm 4870   ran crn 4871    |` cres 4872   "cima 4873   Rel wrel 4875   Fun wfun 5440    Fn wfn 5441   -->wf 5442   -onto->wfo 5444   Univcgru 8657
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-gru 8658
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