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Theorem yonedalem3 15086
Description: Lemma for yoneda 15089. (Contributed by Mario Carneiro, 28-Jan-2017.)
Hypotheses
Ref Expression
yoneda.y  |-  Y  =  (Yon `  C )
yoneda.b  |-  B  =  ( Base `  C
)
yoneda.1  |-  .1.  =  ( Id `  C )
yoneda.o  |-  O  =  (oppCat `  C )
yoneda.s  |-  S  =  ( SetCat `  U )
yoneda.t  |-  T  =  ( SetCat `  V )
yoneda.q  |-  Q  =  ( O FuncCat  S )
yoneda.h  |-  H  =  (HomF
`  Q )
yoneda.r  |-  R  =  ( ( Q  X.c  O
) FuncCat  T )
yoneda.e  |-  E  =  ( O evalF  S )
yoneda.z  |-  Z  =  ( H  o.func  ( ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) )
yoneda.c  |-  ( ph  ->  C  e.  Cat )
yoneda.w  |-  ( ph  ->  V  e.  W )
yoneda.u  |-  ( ph  ->  ran  ( Hom f  `  C ) 
C_  U )
yoneda.v  |-  ( ph  ->  ( ran  ( Hom f  `  Q )  u.  U
)  C_  V )
yoneda.m  |-  M  =  ( f  e.  ( O  Func  S ) ,  x  e.  B  |->  ( a  e.  ( ( ( 1st `  Y
) `  x )
( O Nat  S ) f )  |->  ( ( a `  x ) `
 (  .1.  `  x ) ) ) )
Assertion
Ref Expression
yonedalem3  |-  ( ph  ->  M  e.  ( Z ( ( Q  X.c  O
) Nat  T ) E ) )
Distinct variable groups:    f, a, x,  .1.    C, a, f, x    E, a, f    B, a, f, x    O, a, f, x    S, a, f, x    Q, a, f, x    T, f    ph, a, f, x    Y, a, f, x    Z, a, f, x
Allowed substitution hints:    R( x, f, a)    T( x, a)    U( x, f, a)    E( x)    H( x, f, a)    M( x, f, a)    V( x, f, a)    W( x, f, a)

Proof of Theorem yonedalem3
Dummy variables  g 
y  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 yoneda.m . . . . 5  |-  M  =  ( f  e.  ( O  Func  S ) ,  x  e.  B  |->  ( a  e.  ( ( ( 1st `  Y
) `  x )
( O Nat  S ) f )  |->  ( ( a `  x ) `
 (  .1.  `  x ) ) ) )
2 ovex 6115 . . . . . 6  |-  ( ( ( 1st `  Y
) `  x )
( O Nat  S ) f )  e.  _V
32mptex 5945 . . . . 5  |-  ( a  e.  ( ( ( 1st `  Y ) `
 x ) ( O Nat  S ) f )  |->  ( ( a `
 x ) `  (  .1.  `  x )
) )  e.  _V
41, 3fnmpt2i 6642 . . . 4  |-  M  Fn  ( ( O  Func  S )  X.  B )
54a1i 11 . . 3  |-  ( ph  ->  M  Fn  ( ( O  Func  S )  X.  B ) )
6 yoneda.y . . . . . . . 8  |-  Y  =  (Yon `  C )
7 yoneda.b . . . . . . . 8  |-  B  =  ( Base `  C
)
8 yoneda.1 . . . . . . . 8  |-  .1.  =  ( Id `  C )
9 yoneda.o . . . . . . . 8  |-  O  =  (oppCat `  C )
10 yoneda.s . . . . . . . 8  |-  S  =  ( SetCat `  U )
11 yoneda.t . . . . . . . 8  |-  T  =  ( SetCat `  V )
12 yoneda.q . . . . . . . 8  |-  Q  =  ( O FuncCat  S )
13 yoneda.h . . . . . . . 8  |-  H  =  (HomF
`  Q )
14 yoneda.r . . . . . . . 8  |-  R  =  ( ( Q  X.c  O
) FuncCat  T )
15 yoneda.e . . . . . . . 8  |-  E  =  ( O evalF  S )
16 yoneda.z . . . . . . . 8  |-  Z  =  ( H  o.func  ( ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) )
17 yoneda.c . . . . . . . . 9  |-  ( ph  ->  C  e.  Cat )
1817adantr 462 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  ->  C  e.  Cat )
19 yoneda.w . . . . . . . . 9  |-  ( ph  ->  V  e.  W )
2019adantr 462 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  ->  V  e.  W )
21 yoneda.u . . . . . . . . 9  |-  ( ph  ->  ran  ( Hom f  `  C ) 
C_  U )
2221adantr 462 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  ->  ran  ( Hom f  `  C )  C_  U )
23 yoneda.v . . . . . . . . 9  |-  ( ph  ->  ( ran  ( Hom f  `  Q )  u.  U
)  C_  V )
2423adantr 462 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  -> 
( ran  ( Hom f  `  Q
)  u.  U ) 
C_  V )
25 simprl 750 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  -> 
g  e.  ( O 
Func  S ) )
26 simprr 751 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  -> 
y  e.  B )
276, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 25, 26, 1yonedalem3a 15080 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  -> 
( ( g M y )  =  ( a  e.  ( ( ( 1st `  Y
) `  y )
( O Nat  S ) g )  |->  ( ( a `  y ) `
 (  .1.  `  y ) ) )  /\  ( g M y ) : ( g ( 1st `  Z
) y ) --> ( g ( 1st `  E
) y ) ) )
2827simprd 460 . . . . . 6  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  -> 
( g M y ) : ( g ( 1st `  Z
) y ) --> ( g ( 1st `  E
) y ) )
29 eqid 2441 . . . . . . 7  |-  ( Hom  `  T )  =  ( Hom  `  T )
30 eqid 2441 . . . . . . . . . . 11  |-  ( Q  X.c  O )  =  ( Q  X.c  O )
3112fucbas 14866 . . . . . . . . . . 11  |-  ( O 
Func  S )  =  (
Base `  Q )
329, 7oppcbas 14653 . . . . . . . . . . 11  |-  B  =  ( Base `  O
)
3330, 31, 32xpcbas 14984 . . . . . . . . . 10  |-  ( ( O  Func  S )  X.  B )  =  (
Base `  ( Q  X.c  O ) )
34 eqid 2441 . . . . . . . . . 10  |-  ( Base `  T )  =  (
Base `  T )
35 relfunc 14768 . . . . . . . . . . 11  |-  Rel  (
( Q  X.c  O ) 
Func  T )
366, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 23yonedalem1 15078 . . . . . . . . . . . 12  |-  ( ph  ->  ( Z  e.  ( ( Q  X.c  O ) 
Func  T )  /\  E  e.  ( ( Q  X.c  O
)  Func  T )
) )
3736simpld 456 . . . . . . . . . . 11  |-  ( ph  ->  Z  e.  ( ( Q  X.c  O )  Func  T
) )
38 1st2ndbr 6622 . . . . . . . . . . 11  |-  ( ( Rel  ( ( Q  X.c  O )  Func  T
)  /\  Z  e.  ( ( Q  X.c  O
)  Func  T )
)  ->  ( 1st `  Z ) ( ( Q  X.c  O )  Func  T
) ( 2nd `  Z
) )
3935, 37, 38sylancr 658 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  Z
) ( ( Q  X.c  O )  Func  T
) ( 2nd `  Z
) )
4033, 34, 39funcf1 14772 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  Z
) : ( ( O  Func  S )  X.  B ) --> ( Base `  T ) )
4140fovrnda 6233 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  -> 
( g ( 1st `  Z ) y )  e.  ( Base `  T
) )
4211, 20setcbas 14942 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  ->  V  =  ( Base `  T ) )
4341, 42eleqtrrd 2518 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  -> 
( g ( 1st `  Z ) y )  e.  V )
4436simprd 460 . . . . . . . . . . 11  |-  ( ph  ->  E  e.  ( ( Q  X.c  O )  Func  T
) )
45 1st2ndbr 6622 . . . . . . . . . . 11  |-  ( ( Rel  ( ( Q  X.c  O )  Func  T
)  /\  E  e.  ( ( Q  X.c  O
)  Func  T )
)  ->  ( 1st `  E ) ( ( Q  X.c  O )  Func  T
) ( 2nd `  E
) )
4635, 44, 45sylancr 658 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  E
) ( ( Q  X.c  O )  Func  T
) ( 2nd `  E
) )
4733, 34, 46funcf1 14772 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  E
) : ( ( O  Func  S )  X.  B ) --> ( Base `  T ) )
4847fovrnda 6233 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  -> 
( g ( 1st `  E ) y )  e.  ( Base `  T
) )
4948, 42eleqtrrd 2518 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  -> 
( g ( 1st `  E ) y )  e.  V )
5011, 20, 29, 43, 49elsetchom 14945 . . . . . 6  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  -> 
( ( g M y )  e.  ( ( g ( 1st `  Z ) y ) ( Hom  `  T
) ( g ( 1st `  E ) y ) )  <->  ( g M y ) : ( g ( 1st `  Z ) y ) --> ( g ( 1st `  E ) y ) ) )
5128, 50mpbird 232 . . . . 5  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  -> 
( g M y )  e.  ( ( g ( 1st `  Z
) y ) ( Hom  `  T )
( g ( 1st `  E ) y ) ) )
5251ralrimivva 2806 . . . 4  |-  ( ph  ->  A. g  e.  ( O  Func  S ) A. y  e.  B  ( g M y )  e.  ( ( g ( 1st `  Z
) y ) ( Hom  `  T )
( g ( 1st `  E ) y ) ) )
53 fveq2 5688 . . . . . . 7  |-  ( z  =  <. g ,  y
>.  ->  ( M `  z )  =  ( M `  <. g ,  y >. )
)
54 df-ov 6093 . . . . . . 7  |-  ( g M y )  =  ( M `  <. g ,  y >. )
5553, 54syl6eqr 2491 . . . . . 6  |-  ( z  =  <. g ,  y
>.  ->  ( M `  z )  =  ( g M y ) )
56 fveq2 5688 . . . . . . . 8  |-  ( z  =  <. g ,  y
>.  ->  ( ( 1st `  Z ) `  z
)  =  ( ( 1st `  Z ) `
 <. g ,  y
>. ) )
57 df-ov 6093 . . . . . . . 8  |-  ( g ( 1st `  Z
) y )  =  ( ( 1st `  Z
) `  <. g ,  y >. )
5856, 57syl6eqr 2491 . . . . . . 7  |-  ( z  =  <. g ,  y
>.  ->  ( ( 1st `  Z ) `  z
)  =  ( g ( 1st `  Z
) y ) )
59 fveq2 5688 . . . . . . . 8  |-  ( z  =  <. g ,  y
>.  ->  ( ( 1st `  E ) `  z
)  =  ( ( 1st `  E ) `
 <. g ,  y
>. ) )
60 df-ov 6093 . . . . . . . 8  |-  ( g ( 1st `  E
) y )  =  ( ( 1st `  E
) `  <. g ,  y >. )
6159, 60syl6eqr 2491 . . . . . . 7  |-  ( z  =  <. g ,  y
>.  ->  ( ( 1st `  E ) `  z
)  =  ( g ( 1st `  E
) y ) )
6258, 61oveq12d 6108 . . . . . 6  |-  ( z  =  <. g ,  y
>.  ->  ( ( ( 1st `  Z ) `
 z ) ( Hom  `  T )
( ( 1st `  E
) `  z )
)  =  ( ( g ( 1st `  Z
) y ) ( Hom  `  T )
( g ( 1st `  E ) y ) ) )
6355, 62eleq12d 2509 . . . . 5  |-  ( z  =  <. g ,  y
>.  ->  ( ( M `
 z )  e.  ( ( ( 1st `  Z ) `  z
) ( Hom  `  T
) ( ( 1st `  E ) `  z
) )  <->  ( g M y )  e.  ( ( g ( 1st `  Z ) y ) ( Hom  `  T ) ( g ( 1st `  E
) y ) ) ) )
6463ralxp 4977 . . . 4  |-  ( A. z  e.  ( ( O  Func  S )  X.  B ) ( M `
 z )  e.  ( ( ( 1st `  Z ) `  z
) ( Hom  `  T
) ( ( 1st `  E ) `  z
) )  <->  A. g  e.  ( O  Func  S
) A. y  e.  B  ( g M y )  e.  ( ( g ( 1st `  Z ) y ) ( Hom  `  T
) ( g ( 1st `  E ) y ) ) )
6552, 64sylibr 212 . . 3  |-  ( ph  ->  A. z  e.  ( ( O  Func  S
)  X.  B ) ( M `  z
)  e.  ( ( ( 1st `  Z
) `  z )
( Hom  `  T ) ( ( 1st `  E
) `  z )
) )
66 ovex 6115 . . . . . 6  |-  ( O 
Func  S )  e.  _V
67 fvex 5698 . . . . . . 7  |-  ( Base `  C )  e.  _V
687, 67eqeltri 2511 . . . . . 6  |-  B  e. 
_V
6966, 68mpt2ex 6649 . . . . 5  |-  ( f  e.  ( O  Func  S ) ,  x  e.  B  |->  ( a  e.  ( ( ( 1st `  Y ) `  x
) ( O Nat  S
) f )  |->  ( ( a `  x
) `  (  .1.  `  x ) ) ) )  e.  _V
701, 69eqeltri 2511 . . . 4  |-  M  e. 
_V
7170elixp 7266 . . 3  |-  ( M  e.  X_ z  e.  ( ( O  Func  S
)  X.  B ) ( ( ( 1st `  Z ) `  z
) ( Hom  `  T
) ( ( 1st `  E ) `  z
) )  <->  ( M  Fn  ( ( O  Func  S )  X.  B )  /\  A. z  e.  ( ( O  Func  S )  X.  B ) ( M `  z
)  e.  ( ( ( 1st `  Z
) `  z )
( Hom  `  T ) ( ( 1st `  E
) `  z )
) ) )
725, 65, 71sylanbrc 659 . 2  |-  ( ph  ->  M  e.  X_ z  e.  ( ( O  Func  S )  X.  B ) ( ( ( 1st `  Z ) `  z
) ( Hom  `  T
) ( ( 1st `  E ) `  z
) ) )
7317adantr 462 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  ->  C  e.  Cat )
7419adantr 462 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  ->  V  e.  W )
7521adantr 462 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  ->  ran  ( Hom f  `  C )  C_  U )
7623adantr 462 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( ran  ( Hom f  `  Q
)  u.  U ) 
C_  V )
77 simpr1 989 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
z  e.  ( ( O  Func  S )  X.  B ) )
78 xp1st 6605 . . . . . 6  |-  ( z  e.  ( ( O 
Func  S )  X.  B
)  ->  ( 1st `  z )  e.  ( O  Func  S )
)
7977, 78syl 16 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( 1st `  z
)  e.  ( O 
Func  S ) )
80 xp2nd 6606 . . . . . 6  |-  ( z  e.  ( ( O 
Func  S )  X.  B
)  ->  ( 2nd `  z )  e.  B
)
8177, 80syl 16 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( 2nd `  z
)  e.  B )
82 simpr2 990 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  ->  w  e.  ( ( O  Func  S )  X.  B ) )
83 xp1st 6605 . . . . . 6  |-  ( w  e.  ( ( O 
Func  S )  X.  B
)  ->  ( 1st `  w )  e.  ( O  Func  S )
)
8482, 83syl 16 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( 1st `  w
)  e.  ( O 
Func  S ) )
85 xp2nd 6606 . . . . . 6  |-  ( w  e.  ( ( O 
Func  S )  X.  B
)  ->  ( 2nd `  w )  e.  B
)
8682, 85syl 16 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( 2nd `  w
)  e.  B )
87 simpr3 991 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) )
88 eqid 2441 . . . . . . . . . 10  |-  ( O Nat 
S )  =  ( O Nat  S )
8912, 88fuchom 14867 . . . . . . . . 9  |-  ( O Nat 
S )  =  ( Hom  `  Q )
90 eqid 2441 . . . . . . . . 9  |-  ( Hom  `  O )  =  ( Hom  `  O )
91 eqid 2441 . . . . . . . . 9  |-  ( Hom  `  ( Q  X.c  O ) )  =  ( Hom  `  ( Q  X.c  O ) )
9230, 33, 89, 90, 91, 77, 82xpchom 14986 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( z ( Hom  `  ( Q  X.c  O ) ) w )  =  ( ( ( 1st `  z ) ( O Nat 
S ) ( 1st `  w ) )  X.  ( ( 2nd `  z
) ( Hom  `  O
) ( 2nd `  w
) ) ) )
93 eqid 2441 . . . . . . . . . 10  |-  ( Hom  `  C )  =  ( Hom  `  C )
9493, 9oppchom 14650 . . . . . . . . 9  |-  ( ( 2nd `  z ) ( Hom  `  O
) ( 2nd `  w
) )  =  ( ( 2nd `  w
) ( Hom  `  C
) ( 2nd `  z
) )
9594xpeq2i 4857 . . . . . . . 8  |-  ( ( ( 1st `  z
) ( O Nat  S
) ( 1st `  w
) )  X.  (
( 2nd `  z
) ( Hom  `  O
) ( 2nd `  w
) ) )  =  ( ( ( 1st `  z ) ( O Nat 
S ) ( 1st `  w ) )  X.  ( ( 2nd `  w
) ( Hom  `  C
) ( 2nd `  z
) ) )
9692, 95syl6eq 2489 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( z ( Hom  `  ( Q  X.c  O ) ) w )  =  ( ( ( 1st `  z ) ( O Nat 
S ) ( 1st `  w ) )  X.  ( ( 2nd `  w
) ( Hom  `  C
) ( 2nd `  z
) ) ) )
9787, 96eleqtrd 2517 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
g  e.  ( ( ( 1st `  z
) ( O Nat  S
) ( 1st `  w
) )  X.  (
( 2nd `  w
) ( Hom  `  C
) ( 2nd `  z
) ) ) )
98 xp1st 6605 . . . . . 6  |-  ( g  e.  ( ( ( 1st `  z ) ( O Nat  S ) ( 1st `  w
) )  X.  (
( 2nd `  w
) ( Hom  `  C
) ( 2nd `  z
) ) )  -> 
( 1st `  g
)  e.  ( ( 1st `  z ) ( O Nat  S ) ( 1st `  w
) ) )
9997, 98syl 16 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( 1st `  g
)  e.  ( ( 1st `  z ) ( O Nat  S ) ( 1st `  w
) ) )
100 xp2nd 6606 . . . . . 6  |-  ( g  e.  ( ( ( 1st `  z ) ( O Nat  S ) ( 1st `  w
) )  X.  (
( 2nd `  w
) ( Hom  `  C
) ( 2nd `  z
) ) )  -> 
( 2nd `  g
)  e.  ( ( 2nd `  w ) ( Hom  `  C
) ( 2nd `  z
) ) )
10197, 100syl 16 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( 2nd `  g
)  e.  ( ( 2nd `  w ) ( Hom  `  C
) ( 2nd `  z
) ) )
1026, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 73, 74, 75, 76, 79, 81, 84, 86, 99, 101, 1yonedalem3b 15085 . . . 4  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( ( ( 1st `  w ) M ( 2nd `  w ) ) ( <. (
( 1st `  z
) ( 1st `  Z
) ( 2nd `  z
) ) ,  ( ( 1st `  w
) ( 1st `  Z
) ( 2nd `  w
) ) >. (comp `  T ) ( ( 1st `  w ) ( 1st `  E
) ( 2nd `  w
) ) ) ( ( 1st `  g
) ( <. ( 1st `  z ) ,  ( 2nd `  z
) >. ( 2nd `  Z
) <. ( 1st `  w
) ,  ( 2nd `  w ) >. )
( 2nd `  g
) ) )  =  ( ( ( 1st `  g ) ( <.
( 1st `  z
) ,  ( 2nd `  z ) >. ( 2nd `  E ) <.
( 1st `  w
) ,  ( 2nd `  w ) >. )
( 2nd `  g
) ) ( <.
( ( 1st `  z
) ( 1st `  Z
) ( 2nd `  z
) ) ,  ( ( 1st `  z
) ( 1st `  E
) ( 2nd `  z
) ) >. (comp `  T ) ( ( 1st `  w ) ( 1st `  E
) ( 2nd `  w
) ) ) ( ( 1st `  z
) M ( 2nd `  z ) ) ) )
103 1st2nd2 6612 . . . . . . . . . 10  |-  ( z  e.  ( ( O 
Func  S )  X.  B
)  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
10477, 103syl 16 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
z  =  <. ( 1st `  z ) ,  ( 2nd `  z
) >. )
105104fveq2d 5692 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( ( 1st `  Z
) `  z )  =  ( ( 1st `  Z ) `  <. ( 1st `  z ) ,  ( 2nd `  z
) >. ) )
106 df-ov 6093 . . . . . . . 8  |-  ( ( 1st `  z ) ( 1st `  Z
) ( 2nd `  z
) )  =  ( ( 1st `  Z
) `  <. ( 1st `  z ) ,  ( 2nd `  z )
>. )
107105, 106syl6eqr 2491 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( ( 1st `  Z
) `  z )  =  ( ( 1st `  z ) ( 1st `  Z ) ( 2nd `  z ) ) )
108 1st2nd2 6612 . . . . . . . . . 10  |-  ( w  e.  ( ( O 
Func  S )  X.  B
)  ->  w  =  <. ( 1st `  w
) ,  ( 2nd `  w ) >. )
10982, 108syl 16 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  ->  w  =  <. ( 1st `  w ) ,  ( 2nd `  w )
>. )
110109fveq2d 5692 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( ( 1st `  Z
) `  w )  =  ( ( 1st `  Z ) `  <. ( 1st `  w ) ,  ( 2nd `  w
) >. ) )
111 df-ov 6093 . . . . . . . 8  |-  ( ( 1st `  w ) ( 1st `  Z
) ( 2nd `  w
) )  =  ( ( 1st `  Z
) `  <. ( 1st `  w ) ,  ( 2nd `  w )
>. )
112110, 111syl6eqr 2491 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( ( 1st `  Z
) `  w )  =  ( ( 1st `  w ) ( 1st `  Z ) ( 2nd `  w ) ) )
113107, 112opeq12d 4064 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  ->  <. ( ( 1st `  Z
) `  z ) ,  ( ( 1st `  Z ) `  w
) >.  =  <. (
( 1st `  z
) ( 1st `  Z
) ( 2nd `  z
) ) ,  ( ( 1st `  w
) ( 1st `  Z
) ( 2nd `  w
) ) >. )
114109fveq2d 5692 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( ( 1st `  E
) `  w )  =  ( ( 1st `  E ) `  <. ( 1st `  w ) ,  ( 2nd `  w
) >. ) )
115 df-ov 6093 . . . . . . 7  |-  ( ( 1st `  w ) ( 1st `  E
) ( 2nd `  w
) )  =  ( ( 1st `  E
) `  <. ( 1st `  w ) ,  ( 2nd `  w )
>. )
116114, 115syl6eqr 2491 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( ( 1st `  E
) `  w )  =  ( ( 1st `  w ) ( 1st `  E ) ( 2nd `  w ) ) )
117113, 116oveq12d 6108 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( <. ( ( 1st `  Z ) `  z
) ,  ( ( 1st `  Z ) `
 w ) >.
(comp `  T )
( ( 1st `  E
) `  w )
)  =  ( <.
( ( 1st `  z
) ( 1st `  Z
) ( 2nd `  z
) ) ,  ( ( 1st `  w
) ( 1st `  Z
) ( 2nd `  w
) ) >. (comp `  T ) ( ( 1st `  w ) ( 1st `  E
) ( 2nd `  w
) ) ) )
118109fveq2d 5692 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( M `  w
)  =  ( M `
 <. ( 1st `  w
) ,  ( 2nd `  w ) >. )
)
119 df-ov 6093 . . . . . 6  |-  ( ( 1st `  w ) M ( 2nd `  w
) )  =  ( M `  <. ( 1st `  w ) ,  ( 2nd `  w
) >. )
120118, 119syl6eqr 2491 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( M `  w
)  =  ( ( 1st `  w ) M ( 2nd `  w
) ) )
121104, 109oveq12d 6108 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( z ( 2nd `  Z ) w )  =  ( <. ( 1st `  z ) ,  ( 2nd `  z
) >. ( 2nd `  Z
) <. ( 1st `  w
) ,  ( 2nd `  w ) >. )
)
122 1st2nd2 6612 . . . . . . . 8  |-  ( g  e.  ( ( ( 1st `  z ) ( O Nat  S ) ( 1st `  w
) )  X.  (
( 2nd `  w
) ( Hom  `  C
) ( 2nd `  z
) ) )  -> 
g  =  <. ( 1st `  g ) ,  ( 2nd `  g
) >. )
12397, 122syl 16 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
g  =  <. ( 1st `  g ) ,  ( 2nd `  g
) >. )
124121, 123fveq12d 5694 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( ( z ( 2nd `  Z ) w ) `  g
)  =  ( (
<. ( 1st `  z
) ,  ( 2nd `  z ) >. ( 2nd `  Z ) <.
( 1st `  w
) ,  ( 2nd `  w ) >. ) `  <. ( 1st `  g
) ,  ( 2nd `  g ) >. )
)
125 df-ov 6093 . . . . . 6  |-  ( ( 1st `  g ) ( <. ( 1st `  z
) ,  ( 2nd `  z ) >. ( 2nd `  Z ) <.
( 1st `  w
) ,  ( 2nd `  w ) >. )
( 2nd `  g
) )  =  ( ( <. ( 1st `  z
) ,  ( 2nd `  z ) >. ( 2nd `  Z ) <.
( 1st `  w
) ,  ( 2nd `  w ) >. ) `  <. ( 1st `  g
) ,  ( 2nd `  g ) >. )
126124, 125syl6eqr 2491 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( ( z ( 2nd `  Z ) w ) `  g
)  =  ( ( 1st `  g ) ( <. ( 1st `  z
) ,  ( 2nd `  z ) >. ( 2nd `  Z ) <.
( 1st `  w
) ,  ( 2nd `  w ) >. )
( 2nd `  g
) ) )
127117, 120, 126oveq123d 6111 . . . 4  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( ( M `  w ) ( <.
( ( 1st `  Z
) `  z ) ,  ( ( 1st `  Z ) `  w
) >. (comp `  T
) ( ( 1st `  E ) `  w
) ) ( ( z ( 2nd `  Z
) w ) `  g ) )  =  ( ( ( 1st `  w ) M ( 2nd `  w ) ) ( <. (
( 1st `  z
) ( 1st `  Z
) ( 2nd `  z
) ) ,  ( ( 1st `  w
) ( 1st `  Z
) ( 2nd `  w
) ) >. (comp `  T ) ( ( 1st `  w ) ( 1st `  E
) ( 2nd `  w
) ) ) ( ( 1st `  g
) ( <. ( 1st `  z ) ,  ( 2nd `  z
) >. ( 2nd `  Z
) <. ( 1st `  w
) ,  ( 2nd `  w ) >. )
( 2nd `  g
) ) ) )
128104fveq2d 5692 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( ( 1st `  E
) `  z )  =  ( ( 1st `  E ) `  <. ( 1st `  z ) ,  ( 2nd `  z
) >. ) )
129 df-ov 6093 . . . . . . . 8  |-  ( ( 1st `  z ) ( 1st `  E
) ( 2nd `  z
) )  =  ( ( 1st `  E
) `  <. ( 1st `  z ) ,  ( 2nd `  z )
>. )
130128, 129syl6eqr 2491 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( ( 1st `  E
) `  z )  =  ( ( 1st `  z ) ( 1st `  E ) ( 2nd `  z ) ) )
131107, 130opeq12d 4064 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  ->  <. ( ( 1st `  Z
) `  z ) ,  ( ( 1st `  E ) `  z
) >.  =  <. (
( 1st `  z
) ( 1st `  Z
) ( 2nd `  z
) ) ,  ( ( 1st `  z
) ( 1st `  E
) ( 2nd `  z
) ) >. )
132131, 116oveq12d 6108 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( <. ( ( 1st `  Z ) `  z
) ,  ( ( 1st `  E ) `
 z ) >.
(comp `  T )
( ( 1st `  E
) `  w )
)  =  ( <.
( ( 1st `  z
) ( 1st `  Z
) ( 2nd `  z
) ) ,  ( ( 1st `  z
) ( 1st `  E
) ( 2nd `  z
) ) >. (comp `  T ) ( ( 1st `  w ) ( 1st `  E
) ( 2nd `  w
) ) ) )
133104, 109oveq12d 6108 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( z ( 2nd `  E ) w )  =  ( <. ( 1st `  z ) ,  ( 2nd `  z
) >. ( 2nd `  E
) <. ( 1st `  w
) ,  ( 2nd `  w ) >. )
)
134133, 123fveq12d 5694 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( ( z ( 2nd `  E ) w ) `  g
)  =  ( (
<. ( 1st `  z
) ,  ( 2nd `  z ) >. ( 2nd `  E ) <.
( 1st `  w
) ,  ( 2nd `  w ) >. ) `  <. ( 1st `  g
) ,  ( 2nd `  g ) >. )
)
135 df-ov 6093 . . . . . 6  |-  ( ( 1st `  g ) ( <. ( 1st `  z
) ,  ( 2nd `  z ) >. ( 2nd `  E ) <.
( 1st `  w
) ,  ( 2nd `  w ) >. )
( 2nd `  g
) )  =  ( ( <. ( 1st `  z
) ,  ( 2nd `  z ) >. ( 2nd `  E ) <.
( 1st `  w
) ,  ( 2nd `  w ) >. ) `  <. ( 1st `  g
) ,  ( 2nd `  g ) >. )
136134, 135syl6eqr 2491 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( ( z ( 2nd `  E ) w ) `  g
)  =  ( ( 1st `  g ) ( <. ( 1st `  z
) ,  ( 2nd `  z ) >. ( 2nd `  E ) <.
( 1st `  w
) ,  ( 2nd `  w ) >. )
( 2nd `  g
) ) )
137104fveq2d 5692 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( M `  z
)  =  ( M `
 <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
)
138 df-ov 6093 . . . . . 6  |-  ( ( 1st `  z ) M ( 2nd `  z
) )  =  ( M `  <. ( 1st `  z ) ,  ( 2nd `  z
) >. )
139137, 138syl6eqr 2491 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( M `  z
)  =  ( ( 1st `  z ) M ( 2nd `  z
) ) )
140132, 136, 139oveq123d 6111 . . . 4  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( ( ( z ( 2nd `  E
) w ) `  g ) ( <.
( ( 1st `  Z
) `  z ) ,  ( ( 1st `  E ) `  z
) >. (comp `  T
) ( ( 1st `  E ) `  w
) ) ( M `
 z ) )  =  ( ( ( 1st `  g ) ( <. ( 1st `  z
) ,  ( 2nd `  z ) >. ( 2nd `  E ) <.
( 1st `  w
) ,  ( 2nd `  w ) >. )
( 2nd `  g
) ) ( <.
( ( 1st `  z
) ( 1st `  Z
) ( 2nd `  z
) ) ,  ( ( 1st `  z
) ( 1st `  E
) ( 2nd `  z
) ) >. (comp `  T ) ( ( 1st `  w ) ( 1st `  E
) ( 2nd `  w
) ) ) ( ( 1st `  z
) M ( 2nd `  z ) ) ) )
141102, 127, 1403eqtr4d 2483 . . 3  |-  ( (
ph  /\  ( z  e.  ( ( O  Func  S )  X.  B )  /\  w  e.  ( ( O  Func  S
)  X.  B )  /\  g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ) )  -> 
( ( M `  w ) ( <.
( ( 1st `  Z
) `  z ) ,  ( ( 1st `  Z ) `  w
) >. (comp `  T
) ( ( 1st `  E ) `  w
) ) ( ( z ( 2nd `  Z
) w ) `  g ) )  =  ( ( ( z ( 2nd `  E
) w ) `  g ) ( <.
( ( 1st `  Z
) `  z ) ,  ( ( 1st `  E ) `  z
) >. (comp `  T
) ( ( 1st `  E ) `  w
) ) ( M `
 z ) ) )
142141ralrimivvva 2807 . 2  |-  ( ph  ->  A. z  e.  ( ( O  Func  S
)  X.  B ) A. w  e.  ( ( O  Func  S
)  X.  B ) A. g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ( ( M `
 w ) (
<. ( ( 1st `  Z
) `  z ) ,  ( ( 1st `  Z ) `  w
) >. (comp `  T
) ( ( 1st `  E ) `  w
) ) ( ( z ( 2nd `  Z
) w ) `  g ) )  =  ( ( ( z ( 2nd `  E
) w ) `  g ) ( <.
( ( 1st `  Z
) `  z ) ,  ( ( 1st `  E ) `  z
) >. (comp `  T
) ( ( 1st `  E ) `  w
) ) ( M `
 z ) ) )
143 eqid 2441 . . 3  |-  ( ( Q  X.c  O ) Nat  T )  =  ( ( Q  X.c  O ) Nat  T )
144 eqid 2441 . . 3  |-  (comp `  T )  =  (comp `  T )
145143, 33, 91, 29, 144, 37, 44isnat2 14854 . 2  |-  ( ph  ->  ( M  e.  ( Z ( ( Q  X.c  O ) Nat  T ) E )  <->  ( M  e.  X_ z  e.  ( ( O  Func  S
)  X.  B ) ( ( ( 1st `  Z ) `  z
) ( Hom  `  T
) ( ( 1st `  E ) `  z
) )  /\  A. z  e.  ( ( O  Func  S )  X.  B ) A. w  e.  ( ( O  Func  S )  X.  B ) A. g  e.  ( z ( Hom  `  ( Q  X.c  O ) ) w ) ( ( M `
 w ) (
<. ( ( 1st `  Z
) `  z ) ,  ( ( 1st `  Z ) `  w
) >. (comp `  T
) ( ( 1st `  E ) `  w
) ) ( ( z ( 2nd `  Z
) w ) `  g ) )  =  ( ( ( z ( 2nd `  E
) w ) `  g ) ( <.
( ( 1st `  Z
) `  z ) ,  ( ( 1st `  E ) `  z
) >. (comp `  T
) ( ( 1st `  E ) `  w
) ) ( M `
 z ) ) ) ) )
14672, 142, 145mpbir2and 908 1  |-  ( ph  ->  M  e.  ( Z ( ( Q  X.c  O
) Nat  T ) E ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   A.wral 2713   _Vcvv 2970    u. cun 3323    C_ wss 3325   <.cop 3880   class class class wbr 4289    e. cmpt 4347    X. cxp 4834   ran crn 4837   Rel wrel 4841    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092   1stc1st 6574   2ndc2nd 6575  tpos ctpos 6743   X_cixp 7259   Basecbs 14170   Hom chom 14245  compcco 14246   Catccat 14598   Idccid 14599   Hom f chomf 14600  oppCatcoppc 14646    Func cfunc 14760    o.func ccofu 14762   Nat cnat 14847   FuncCat cfuc 14848   SetCatcsetc 14939    X.c cxpc 14974    1stF c1stf 14975    2ndF c2ndf 14976   ⟨,⟩F cprf 14977   evalF cevlf 15015  HomFchof 15054  Yoncyon 15055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-tpos 6744  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-fz 11434  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-hom 14258  df-cco 14259  df-cat 14602  df-cid 14603  df-homf 14604  df-comf 14605  df-oppc 14647  df-ssc 14719  df-resc 14720  df-subc 14721  df-func 14764  df-cofu 14766  df-nat 14849  df-fuc 14850  df-setc 14940  df-xpc 14978  df-1stf 14979  df-2ndf 14980  df-prf 14981  df-evlf 15019  df-curf 15020  df-hof 15056  df-yon 15057
This theorem is referenced by:  yonedainv  15087
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