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Theorem wunfunc 14814
Description: A weak universe is closed under the functor set operation. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
wunfunc.1  |-  ( ph  ->  U  e. WUni )
wunfunc.2  |-  ( ph  ->  C  e.  U )
wunfunc.3  |-  ( ph  ->  D  e.  U )
Assertion
Ref Expression
wunfunc  |-  ( ph  ->  ( C  Func  D
)  e.  U )

Proof of Theorem wunfunc
Dummy variables  f 
g  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wunfunc.1 . 2  |-  ( ph  ->  U  e. WUni )
2 df-base 14184 . . . . 5  |-  Base  = Slot  1
3 wunfunc.3 . . . . 5  |-  ( ph  ->  D  e.  U )
42, 1, 3wunstr 14198 . . . 4  |-  ( ph  ->  ( Base `  D
)  e.  U )
5 wunfunc.2 . . . . 5  |-  ( ph  ->  C  e.  U )
62, 1, 5wunstr 14198 . . . 4  |-  ( ph  ->  ( Base `  C
)  e.  U )
71, 4, 6wunmap 8898 . . 3  |-  ( ph  ->  ( ( Base `  D
)  ^m  ( Base `  C ) )  e.  U )
8 df-hom 14267 . . . . . . . . 9  |-  Hom  = Slot ; 1 4
98, 1, 5wunstr 14198 . . . . . . . 8  |-  ( ph  ->  ( Hom  `  C
)  e.  U )
101, 9wunrn 8901 . . . . . . 7  |-  ( ph  ->  ran  ( Hom  `  C
)  e.  U )
111, 10wununi 8878 . . . . . 6  |-  ( ph  ->  U. ran  ( Hom  `  C )  e.  U
)
128, 1, 3wunstr 14198 . . . . . . . 8  |-  ( ph  ->  ( Hom  `  D
)  e.  U )
131, 12wunrn 8901 . . . . . . 7  |-  ( ph  ->  ran  ( Hom  `  D
)  e.  U )
141, 13wununi 8878 . . . . . 6  |-  ( ph  ->  U. ran  ( Hom  `  D )  e.  U
)
151, 11, 14wunxp 8896 . . . . 5  |-  ( ph  ->  ( U. ran  ( Hom  `  C )  X. 
U. ran  ( Hom  `  D ) )  e.  U )
161, 15wunpw 8879 . . . 4  |-  ( ph  ->  ~P ( U. ran  ( Hom  `  C )  X.  U. ran  ( Hom  `  D ) )  e.  U )
171, 6, 6wunxp 8896 . . . 4  |-  ( ph  ->  ( ( Base `  C
)  X.  ( Base `  C ) )  e.  U )
181, 16, 17wunmap 8898 . . 3  |-  ( ph  ->  ( ~P ( U. ran  ( Hom  `  C
)  X.  U. ran  ( Hom  `  D )
)  ^m  ( ( Base `  C )  X.  ( Base `  C
) ) )  e.  U )
191, 7, 18wunxp 8896 . 2  |-  ( ph  ->  ( ( ( Base `  D )  ^m  ( Base `  C ) )  X.  ( ~P ( U. ran  ( Hom  `  C
)  X.  U. ran  ( Hom  `  D )
)  ^m  ( ( Base `  C )  X.  ( Base `  C
) ) ) )  e.  U )
20 relfunc 14777 . . . 4  |-  Rel  ( C  Func  D )
2120a1i 11 . . 3  |-  ( ph  ->  Rel  ( C  Func  D ) )
22 df-br 4298 . . . 4  |-  ( f ( C  Func  D
) g  <->  <. f ,  g >.  e.  ( C  Func  D ) )
23 eqid 2443 . . . . . . . 8  |-  ( Base `  C )  =  (
Base `  C )
24 eqid 2443 . . . . . . . 8  |-  ( Base `  D )  =  (
Base `  D )
25 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  f ( C  Func  D ) g )  ->  f ( C  Func  D ) g )
2623, 24, 25funcf1 14781 . . . . . . 7  |-  ( (
ph  /\  f ( C  Func  D ) g )  ->  f :
( Base `  C ) --> ( Base `  D )
)
27 fvex 5706 . . . . . . . 8  |-  ( Base `  D )  e.  _V
28 fvex 5706 . . . . . . . 8  |-  ( Base `  C )  e.  _V
2927, 28elmap 7246 . . . . . . 7  |-  ( f  e.  ( ( Base `  D )  ^m  ( Base `  C ) )  <-> 
f : ( Base `  C ) --> ( Base `  D ) )
3026, 29sylibr 212 . . . . . 6  |-  ( (
ph  /\  f ( C  Func  D ) g )  ->  f  e.  ( ( Base `  D
)  ^m  ( Base `  C ) ) )
31 mapsspw 7253 . . . . . . . . . . 11  |-  ( ( ( f `  ( 1st `  z ) ) ( Hom  `  D
) ( f `  ( 2nd `  z ) ) )  ^m  (
( Hom  `  C ) `
 z ) ) 
C_  ~P ( ( ( Hom  `  C ) `  z )  X.  (
( f `  ( 1st `  z ) ) ( Hom  `  D
) ( f `  ( 2nd `  z ) ) ) )
32 fvssunirn 5718 . . . . . . . . . . . . 13  |-  ( ( Hom  `  C ) `  z )  C_  U. ran  ( Hom  `  C )
33 ovssunirn 6122 . . . . . . . . . . . . 13  |-  ( ( f `  ( 1st `  z ) ) ( Hom  `  D )
( f `  ( 2nd `  z ) ) )  C_  U. ran  ( Hom  `  D )
34 xpss12 4950 . . . . . . . . . . . . 13  |-  ( ( ( ( Hom  `  C
) `  z )  C_ 
U. ran  ( Hom  `  C )  /\  (
( f `  ( 1st `  z ) ) ( Hom  `  D
) ( f `  ( 2nd `  z ) ) )  C_  U. ran  ( Hom  `  D )
)  ->  ( (
( Hom  `  C ) `
 z )  X.  ( ( f `  ( 1st `  z ) ) ( Hom  `  D
) ( f `  ( 2nd `  z ) ) ) )  C_  ( U. ran  ( Hom  `  C )  X.  U. ran  ( Hom  `  D
) ) )
3532, 33, 34mp2an 672 . . . . . . . . . . . 12  |-  ( ( ( Hom  `  C
) `  z )  X.  ( ( f `  ( 1st `  z ) ) ( Hom  `  D
) ( f `  ( 2nd `  z ) ) ) )  C_  ( U. ran  ( Hom  `  C )  X.  U. ran  ( Hom  `  D
) )
36 sspwb 4546 . . . . . . . . . . . 12  |-  ( ( ( ( Hom  `  C
) `  z )  X.  ( ( f `  ( 1st `  z ) ) ( Hom  `  D
) ( f `  ( 2nd `  z ) ) ) )  C_  ( U. ran  ( Hom  `  C )  X.  U. ran  ( Hom  `  D
) )  <->  ~P (
( ( Hom  `  C
) `  z )  X.  ( ( f `  ( 1st `  z ) ) ( Hom  `  D
) ( f `  ( 2nd `  z ) ) ) )  C_  ~P ( U. ran  ( Hom  `  C )  X. 
U. ran  ( Hom  `  D ) ) )
3735, 36mpbi 208 . . . . . . . . . . 11  |-  ~P (
( ( Hom  `  C
) `  z )  X.  ( ( f `  ( 1st `  z ) ) ( Hom  `  D
) ( f `  ( 2nd `  z ) ) ) )  C_  ~P ( U. ran  ( Hom  `  C )  X. 
U. ran  ( Hom  `  D ) )
3831, 37sstri 3370 . . . . . . . . . 10  |-  ( ( ( f `  ( 1st `  z ) ) ( Hom  `  D
) ( f `  ( 2nd `  z ) ) )  ^m  (
( Hom  `  C ) `
 z ) ) 
C_  ~P ( U. ran  ( Hom  `  C )  X.  U. ran  ( Hom  `  D ) )
3938rgenw 2788 . . . . . . . . 9  |-  A. z  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ( ( ( f `  ( 1st `  z ) ) ( Hom  `  D
) ( f `  ( 2nd `  z ) ) )  ^m  (
( Hom  `  C ) `
 z ) ) 
C_  ~P ( U. ran  ( Hom  `  C )  X.  U. ran  ( Hom  `  D ) )
40 ss2ixp 7281 . . . . . . . . 9  |-  ( A. z  e.  ( ( Base `  C )  X.  ( Base `  C
) ) ( ( ( f `  ( 1st `  z ) ) ( Hom  `  D
) ( f `  ( 2nd `  z ) ) )  ^m  (
( Hom  `  C ) `
 z ) ) 
C_  ~P ( U. ran  ( Hom  `  C )  X.  U. ran  ( Hom  `  D ) )  ->  X_ z  e.  ( (
Base `  C )  X.  ( Base `  C
) ) ( ( ( f `  ( 1st `  z ) ) ( Hom  `  D
) ( f `  ( 2nd `  z ) ) )  ^m  (
( Hom  `  C ) `
 z ) ) 
C_  X_ z  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ~P ( U. ran  ( Hom  `  C )  X. 
U. ran  ( Hom  `  D ) ) )
4139, 40ax-mp 5 . . . . . . . 8  |-  X_ z  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ( ( ( f `  ( 1st `  z ) ) ( Hom  `  D
) ( f `  ( 2nd `  z ) ) )  ^m  (
( Hom  `  C ) `
 z ) ) 
C_  X_ z  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ~P ( U. ran  ( Hom  `  C )  X. 
U. ran  ( Hom  `  D ) )
4228, 28xpex 6513 . . . . . . . . 9  |-  ( (
Base `  C )  X.  ( Base `  C
) )  e.  _V
43 fvex 5706 . . . . . . . . . . . . 13  |-  ( Hom  `  C )  e.  _V
4443rnex 6517 . . . . . . . . . . . 12  |-  ran  ( Hom  `  C )  e. 
_V
4544uniex 6381 . . . . . . . . . . 11  |-  U. ran  ( Hom  `  C )  e.  _V
46 fvex 5706 . . . . . . . . . . . . 13  |-  ( Hom  `  D )  e.  _V
4746rnex 6517 . . . . . . . . . . . 12  |-  ran  ( Hom  `  D )  e. 
_V
4847uniex 6381 . . . . . . . . . . 11  |-  U. ran  ( Hom  `  D )  e.  _V
4945, 48xpex 6513 . . . . . . . . . 10  |-  ( U. ran  ( Hom  `  C
)  X.  U. ran  ( Hom  `  D )
)  e.  _V
5049pwex 4480 . . . . . . . . 9  |-  ~P ( U. ran  ( Hom  `  C
)  X.  U. ran  ( Hom  `  D )
)  e.  _V
5142, 50ixpconst 7278 . . . . . . . 8  |-  X_ z  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ~P ( U. ran  ( Hom  `  C )  X. 
U. ran  ( Hom  `  D ) )  =  ( ~P ( U. ran  ( Hom  `  C
)  X.  U. ran  ( Hom  `  D )
)  ^m  ( ( Base `  C )  X.  ( Base `  C
) ) )
5241, 51sseqtri 3393 . . . . . . 7  |-  X_ z  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ( ( ( f `  ( 1st `  z ) ) ( Hom  `  D
) ( f `  ( 2nd `  z ) ) )  ^m  (
( Hom  `  C ) `
 z ) ) 
C_  ( ~P ( U. ran  ( Hom  `  C
)  X.  U. ran  ( Hom  `  D )
)  ^m  ( ( Base `  C )  X.  ( Base `  C
) ) )
53 eqid 2443 . . . . . . . 8  |-  ( Hom  `  C )  =  ( Hom  `  C )
54 eqid 2443 . . . . . . . 8  |-  ( Hom  `  D )  =  ( Hom  `  D )
5523, 53, 54, 25funcixp 14782 . . . . . . 7  |-  ( (
ph  /\  f ( C  Func  D ) g )  ->  g  e.  X_ z  e.  ( (
Base `  C )  X.  ( Base `  C
) ) ( ( ( f `  ( 1st `  z ) ) ( Hom  `  D
) ( f `  ( 2nd `  z ) ) )  ^m  (
( Hom  `  C ) `
 z ) ) )
5652, 55sseldi 3359 . . . . . 6  |-  ( (
ph  /\  f ( C  Func  D ) g )  ->  g  e.  ( ~P ( U. ran  ( Hom  `  C )  X.  U. ran  ( Hom  `  D ) )  ^m  ( ( Base `  C
)  X.  ( Base `  C ) ) ) )
57 opelxpi 4876 . . . . . 6  |-  ( ( f  e.  ( (
Base `  D )  ^m  ( Base `  C
) )  /\  g  e.  ( ~P ( U. ran  ( Hom  `  C
)  X.  U. ran  ( Hom  `  D )
)  ^m  ( ( Base `  C )  X.  ( Base `  C
) ) ) )  ->  <. f ,  g
>.  e.  ( ( (
Base `  D )  ^m  ( Base `  C
) )  X.  ( ~P ( U. ran  ( Hom  `  C )  X. 
U. ran  ( Hom  `  D ) )  ^m  ( ( Base `  C
)  X.  ( Base `  C ) ) ) ) )
5830, 56, 57syl2anc 661 . . . . 5  |-  ( (
ph  /\  f ( C  Func  D ) g )  ->  <. f ,  g >.  e.  (
( ( Base `  D
)  ^m  ( Base `  C ) )  X.  ( ~P ( U. ran  ( Hom  `  C
)  X.  U. ran  ( Hom  `  D )
)  ^m  ( ( Base `  C )  X.  ( Base `  C
) ) ) ) )
5958ex 434 . . . 4  |-  ( ph  ->  ( f ( C 
Func  D ) g  ->  <. f ,  g >.  e.  ( ( ( Base `  D )  ^m  ( Base `  C ) )  X.  ( ~P ( U. ran  ( Hom  `  C
)  X.  U. ran  ( Hom  `  D )
)  ^m  ( ( Base `  C )  X.  ( Base `  C
) ) ) ) ) )
6022, 59syl5bir 218 . . 3  |-  ( ph  ->  ( <. f ,  g
>.  e.  ( C  Func  D )  ->  <. f ,  g >.  e.  (
( ( Base `  D
)  ^m  ( Base `  C ) )  X.  ( ~P ( U. ran  ( Hom  `  C
)  X.  U. ran  ( Hom  `  D )
)  ^m  ( ( Base `  C )  X.  ( Base `  C
) ) ) ) ) )
6121, 60relssdv 4937 . 2  |-  ( ph  ->  ( C  Func  D
)  C_  ( (
( Base `  D )  ^m  ( Base `  C
) )  X.  ( ~P ( U. ran  ( Hom  `  C )  X. 
U. ran  ( Hom  `  D ) )  ^m  ( ( Base `  C
)  X.  ( Base `  C ) ) ) ) )
621, 19, 61wunss 8884 1  |-  ( ph  ->  ( C  Func  D
)  e.  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1756   A.wral 2720    C_ wss 3333   ~Pcpw 3865   <.cop 3888   U.cuni 4096   class class class wbr 4297    X. cxp 4843   ran crn 4846   Rel wrel 4850   -->wf 5419   ` cfv 5423  (class class class)co 6096   1stc1st 6580   2ndc2nd 6581    ^m cmap 7219   X_cixp 7268  WUnicwun 8872   1c1 9288   4c4 10378  ;cdc 10760   Basecbs 14179   Hom chom 14254    Func cfunc 14769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-map 7221  df-pm 7222  df-ixp 7269  df-wun 8874  df-slot 14183  df-base 14184  df-hom 14267  df-func 14773
This theorem is referenced by:  wunnat  14871  catcfuccl  14982
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