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Theorem wunfunc 15510
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 14844 . . . . 5  |-  Base  = Slot  1
3 wunfunc.3 . . . . 5  |-  ( ph  ->  D  e.  U )
42, 1, 3wunstr 14858 . . . 4  |-  ( ph  ->  ( Base `  D
)  e.  U )
5 wunfunc.2 . . . . 5  |-  ( ph  ->  C  e.  U )
62, 1, 5wunstr 14858 . . . 4  |-  ( ph  ->  ( Base `  C
)  e.  U )
71, 4, 6wunmap 9133 . . 3  |-  ( ph  ->  ( ( Base `  D
)  ^m  ( Base `  C ) )  e.  U )
8 df-hom 14931 . . . . . . . . 9  |-  Hom  = Slot ; 1 4
98, 1, 5wunstr 14858 . . . . . . . 8  |-  ( ph  ->  ( Hom  `  C
)  e.  U )
101, 9wunrn 9136 . . . . . . 7  |-  ( ph  ->  ran  ( Hom  `  C
)  e.  U )
111, 10wununi 9113 . . . . . 6  |-  ( ph  ->  U. ran  ( Hom  `  C )  e.  U
)
128, 1, 3wunstr 14858 . . . . . . . 8  |-  ( ph  ->  ( Hom  `  D
)  e.  U )
131, 12wunrn 9136 . . . . . . 7  |-  ( ph  ->  ran  ( Hom  `  D
)  e.  U )
141, 13wununi 9113 . . . . . 6  |-  ( ph  ->  U. ran  ( Hom  `  D )  e.  U
)
151, 11, 14wunxp 9131 . . . . 5  |-  ( ph  ->  ( U. ran  ( Hom  `  C )  X. 
U. ran  ( Hom  `  D ) )  e.  U )
161, 15wunpw 9114 . . . 4  |-  ( ph  ->  ~P ( U. ran  ( Hom  `  C )  X.  U. ran  ( Hom  `  D ) )  e.  U )
171, 6, 6wunxp 9131 . . . 4  |-  ( ph  ->  ( ( Base `  C
)  X.  ( Base `  C ) )  e.  U )
181, 16, 17wunmap 9133 . . 3  |-  ( ph  ->  ( ~P ( U. ran  ( Hom  `  C
)  X.  U. ran  ( Hom  `  D )
)  ^m  ( ( Base `  C )  X.  ( Base `  C
) ) )  e.  U )
191, 7, 18wunxp 9131 . 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 15473 . . . 4  |-  Rel  ( C  Func  D )
2120a1i 11 . . 3  |-  ( ph  ->  Rel  ( C  Func  D ) )
22 df-br 4395 . . . 4  |-  ( f ( C  Func  D
) g  <->  <. f ,  g >.  e.  ( C  Func  D ) )
23 eqid 2402 . . . . . . . 8  |-  ( Base `  C )  =  (
Base `  C )
24 eqid 2402 . . . . . . . 8  |-  ( Base `  D )  =  (
Base `  D )
25 simpr 459 . . . . . . . 8  |-  ( (
ph  /\  f ( C  Func  D ) g )  ->  f ( C  Func  D ) g )
2623, 24, 25funcf1 15477 . . . . . . 7  |-  ( (
ph  /\  f ( C  Func  D ) g )  ->  f :
( Base `  C ) --> ( Base `  D )
)
27 fvex 5858 . . . . . . . 8  |-  ( Base `  D )  e.  _V
28 fvex 5858 . . . . . . . 8  |-  ( Base `  C )  e.  _V
2927, 28elmap 7484 . . . . . . 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 7491 . . . . . . . . . . 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 5871 . . . . . . . . . . . . 13  |-  ( ( Hom  `  C ) `  z )  C_  U. ran  ( Hom  `  C )
33 ovssunirn 6306 . . . . . . . . . . . . 13  |-  ( ( f `  ( 1st `  z ) ) ( Hom  `  D )
( f `  ( 2nd `  z ) ) )  C_  U. ran  ( Hom  `  D )
34 xpss12 4928 . . . . . . . . . . . . 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 670 . . . . . . . . . . . 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 4639 . . . . . . . . . . . 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 3450 . . . . . . . . . 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 2764 . . . . . . . . 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 7519 . . . . . . . . 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 6585 . . . . . . . . 9  |-  ( (
Base `  C )  X.  ( Base `  C
) )  e.  _V
43 fvex 5858 . . . . . . . . . . . . 13  |-  ( Hom  `  C )  e.  _V
4443rnex 6717 . . . . . . . . . . . 12  |-  ran  ( Hom  `  C )  e. 
_V
4544uniex 6577 . . . . . . . . . . 11  |-  U. ran  ( Hom  `  C )  e.  _V
46 fvex 5858 . . . . . . . . . . . . 13  |-  ( Hom  `  D )  e.  _V
4746rnex 6717 . . . . . . . . . . . 12  |-  ran  ( Hom  `  D )  e. 
_V
4847uniex 6577 . . . . . . . . . . 11  |-  U. ran  ( Hom  `  D )  e.  _V
4945, 48xpex 6585 . . . . . . . . . 10  |-  ( U. ran  ( Hom  `  C
)  X.  U. ran  ( Hom  `  D )
)  e.  _V
5049pwex 4576 . . . . . . . . 9  |-  ~P ( U. ran  ( Hom  `  C
)  X.  U. ran  ( Hom  `  D )
)  e.  _V
5142, 50ixpconst 7516 . . . . . . . 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 3473 . . . . . . 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 2402 . . . . . . . 8  |-  ( Hom  `  C )  =  ( Hom  `  C )
54 eqid 2402 . . . . . . . 8  |-  ( Hom  `  D )  =  ( Hom  `  D )
5523, 53, 54, 25funcixp 15478 . . . . . . 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 3439 . . . . . 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 4854 . . . . . 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 659 . . . . 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 432 . . . 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 4915 . 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 9119 1  |-  ( ph  ->  ( C  Func  D
)  e.  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1842   A.wral 2753    C_ wss 3413   ~Pcpw 3954   <.cop 3977   U.cuni 4190   class class class wbr 4394    X. cxp 4820   ran crn 4823   Rel wrel 4827   -->wf 5564   ` cfv 5568  (class class class)co 6277   1stc1st 6781   2ndc2nd 6782    ^m cmap 7456   X_cixp 7506  WUnicwun 9107   1c1 9522   4c4 10627  ;cdc 11018   Basecbs 14839   Hom chom 14918    Func cfunc 15465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-map 7458  df-pm 7459  df-ixp 7507  df-wun 9109  df-slot 14843  df-base 14844  df-hom 14931  df-func 15469
This theorem is referenced by:  wunnat  15567  catcfuccl  15710
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