MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wunfunc Structured version   Unicode version

Theorem wunfunc 15315
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 14649 . . . . 5  |-  Base  = Slot  1
3 wunfunc.3 . . . . 5  |-  ( ph  ->  D  e.  U )
42, 1, 3wunstr 14663 . . . 4  |-  ( ph  ->  ( Base `  D
)  e.  U )
5 wunfunc.2 . . . . 5  |-  ( ph  ->  C  e.  U )
62, 1, 5wunstr 14663 . . . 4  |-  ( ph  ->  ( Base `  C
)  e.  U )
71, 4, 6wunmap 9121 . . 3  |-  ( ph  ->  ( ( Base `  D
)  ^m  ( Base `  C ) )  e.  U )
8 df-hom 14736 . . . . . . . . 9  |-  Hom  = Slot ; 1 4
98, 1, 5wunstr 14663 . . . . . . . 8  |-  ( ph  ->  ( Hom  `  C
)  e.  U )
101, 9wunrn 9124 . . . . . . 7  |-  ( ph  ->  ran  ( Hom  `  C
)  e.  U )
111, 10wununi 9101 . . . . . 6  |-  ( ph  ->  U. ran  ( Hom  `  C )  e.  U
)
128, 1, 3wunstr 14663 . . . . . . . 8  |-  ( ph  ->  ( Hom  `  D
)  e.  U )
131, 12wunrn 9124 . . . . . . 7  |-  ( ph  ->  ran  ( Hom  `  D
)  e.  U )
141, 13wununi 9101 . . . . . 6  |-  ( ph  ->  U. ran  ( Hom  `  D )  e.  U
)
151, 11, 14wunxp 9119 . . . . 5  |-  ( ph  ->  ( U. ran  ( Hom  `  C )  X. 
U. ran  ( Hom  `  D ) )  e.  U )
161, 15wunpw 9102 . . . 4  |-  ( ph  ->  ~P ( U. ran  ( Hom  `  C )  X.  U. ran  ( Hom  `  D ) )  e.  U )
171, 6, 6wunxp 9119 . . . 4  |-  ( ph  ->  ( ( Base `  C
)  X.  ( Base `  C ) )  e.  U )
181, 16, 17wunmap 9121 . . 3  |-  ( ph  ->  ( ~P ( U. ran  ( Hom  `  C
)  X.  U. ran  ( Hom  `  D )
)  ^m  ( ( Base `  C )  X.  ( Base `  C
) ) )  e.  U )
191, 7, 18wunxp 9119 . 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 15278 . . . 4  |-  Rel  ( C  Func  D )
2120a1i 11 . . 3  |-  ( ph  ->  Rel  ( C  Func  D ) )
22 df-br 4457 . . . 4  |-  ( f ( C  Func  D
) g  <->  <. f ,  g >.  e.  ( C  Func  D ) )
23 eqid 2457 . . . . . . . 8  |-  ( Base `  C )  =  (
Base `  C )
24 eqid 2457 . . . . . . . 8  |-  ( Base `  D )  =  (
Base `  D )
25 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  f ( C  Func  D ) g )  ->  f ( C  Func  D ) g )
2623, 24, 25funcf1 15282 . . . . . . 7  |-  ( (
ph  /\  f ( C  Func  D ) g )  ->  f :
( Base `  C ) --> ( Base `  D )
)
27 fvex 5882 . . . . . . . 8  |-  ( Base `  D )  e.  _V
28 fvex 5882 . . . . . . . 8  |-  ( Base `  C )  e.  _V
2927, 28elmap 7466 . . . . . . 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 7473 . . . . . . . . . . 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 5895 . . . . . . . . . . . . 13  |-  ( ( Hom  `  C ) `  z )  C_  U. ran  ( Hom  `  C )
33 ovssunirn 6325 . . . . . . . . . . . . 13  |-  ( ( f `  ( 1st `  z ) ) ( Hom  `  D )
( f `  ( 2nd `  z ) ) )  C_  U. ran  ( Hom  `  D )
34 xpss12 5117 . . . . . . . . . . . . 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 4705 . . . . . . . . . . . 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 3508 . . . . . . . . . 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 2818 . . . . . . . . 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 7501 . . . . . . . . 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 6603 . . . . . . . . 9  |-  ( (
Base `  C )  X.  ( Base `  C
) )  e.  _V
43 fvex 5882 . . . . . . . . . . . . 13  |-  ( Hom  `  C )  e.  _V
4443rnex 6733 . . . . . . . . . . . 12  |-  ran  ( Hom  `  C )  e. 
_V
4544uniex 6595 . . . . . . . . . . 11  |-  U. ran  ( Hom  `  C )  e.  _V
46 fvex 5882 . . . . . . . . . . . . 13  |-  ( Hom  `  D )  e.  _V
4746rnex 6733 . . . . . . . . . . . 12  |-  ran  ( Hom  `  D )  e. 
_V
4847uniex 6595 . . . . . . . . . . 11  |-  U. ran  ( Hom  `  D )  e.  _V
4945, 48xpex 6603 . . . . . . . . . 10  |-  ( U. ran  ( Hom  `  C
)  X.  U. ran  ( Hom  `  D )
)  e.  _V
5049pwex 4639 . . . . . . . . 9  |-  ~P ( U. ran  ( Hom  `  C
)  X.  U. ran  ( Hom  `  D )
)  e.  _V
5142, 50ixpconst 7498 . . . . . . . 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 3531 . . . . . . 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 2457 . . . . . . . 8  |-  ( Hom  `  C )  =  ( Hom  `  C )
54 eqid 2457 . . . . . . . 8  |-  ( Hom  `  D )  =  ( Hom  `  D )
5523, 53, 54, 25funcixp 15283 . . . . . . 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 3497 . . . . . 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 5040 . . . . . 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 5104 . 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 9107 1  |-  ( ph  ->  ( C  Func  D
)  e.  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1819   A.wral 2807    C_ wss 3471   ~Pcpw 4015   <.cop 4038   U.cuni 4251   class class class wbr 4456    X. cxp 5006   ran crn 5009   Rel wrel 5013   -->wf 5590   ` cfv 5594  (class class class)co 6296   1stc1st 6797   2ndc2nd 6798    ^m cmap 7438   X_cixp 7488  WUnicwun 9095   1c1 9510   4c4 10608  ;cdc 11000   Basecbs 14644   Hom chom 14723    Func cfunc 15270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-map 7440  df-pm 7441  df-ixp 7489  df-wun 9097  df-slot 14648  df-base 14649  df-hom 14736  df-func 15274
This theorem is referenced by:  wunnat  15372  catcfuccl  15515
  Copyright terms: Public domain W3C validator