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Theorem yonedalem3 16243
Description: Lemma for yoneda 16246. (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 6336 . . . . . 6  |-  ( ( ( 1st `  Y
) `  x )
( O Nat  S ) f )  e.  _V
32mptex 6152 . . . . 5  |-  ( a  e.  ( ( ( 1st `  Y ) `
 x ) ( O Nat  S ) f )  |->  ( ( a `
 x ) `  (  .1.  `  x )
) )  e.  _V
41, 3fnmpt2i 6881 . . . 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 472 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  ->  C  e.  Cat )
19 yoneda.w . . . . . . . . 9  |-  ( ph  ->  V  e.  W )
2019adantr 472 . . . . . . . 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 472 . . . . . . . 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 472 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  -> 
( ran  ( Hom f  `  Q
)  u.  U ) 
C_  V )
25 simprl 772 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  -> 
g  e.  ( O 
Func  S ) )
26 simprr 774 . . . . . . . 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 16237 . . . . . . 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 470 . . . . . 6  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  -> 
( g M y ) : ( g ( 1st `  Z
) y ) --> ( g ( 1st `  E
) y ) )
29 eqid 2471 . . . . . . 7  |-  ( Hom  `  T )  =  ( Hom  `  T )
30 eqid 2471 . . . . . . . . . . 11  |-  ( Q  X.c  O )  =  ( Q  X.c  O )
3112fucbas 15943 . . . . . . . . . . 11  |-  ( O 
Func  S )  =  (
Base `  Q )
329, 7oppcbas 15701 . . . . . . . . . . 11  |-  B  =  ( Base `  O
)
3330, 31, 32xpcbas 16141 . . . . . . . . . 10  |-  ( ( O  Func  S )  X.  B )  =  (
Base `  ( Q  X.c  O ) )
34 eqid 2471 . . . . . . . . . 10  |-  ( Base `  T )  =  (
Base `  T )
35 relfunc 15845 . . . . . . . . . . 11  |-  Rel  (
( Q  X.c  O ) 
Func  T )
366, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 23yonedalem1 16235 . . . . . . . . . . . 12  |-  ( ph  ->  ( Z  e.  ( ( Q  X.c  O ) 
Func  T )  /\  E  e.  ( ( Q  X.c  O
)  Func  T )
) )
3736simpld 466 . . . . . . . . . . 11  |-  ( ph  ->  Z  e.  ( ( Q  X.c  O )  Func  T
) )
38 1st2ndbr 6861 . . . . . . . . . . 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 676 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  Z
) ( ( Q  X.c  O )  Func  T
) ( 2nd `  Z
) )
4033, 34, 39funcf1 15849 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  Z
) : ( ( O  Func  S )  X.  B ) --> ( Base `  T ) )
4140fovrnda 6459 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  -> 
( g ( 1st `  Z ) y )  e.  ( Base `  T
) )
4211, 20setcbas 16051 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  ->  V  =  ( Base `  T ) )
4341, 42eleqtrrd 2552 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  -> 
( g ( 1st `  Z ) y )  e.  V )
4436simprd 470 . . . . . . . . . . 11  |-  ( ph  ->  E  e.  ( ( Q  X.c  O )  Func  T
) )
45 1st2ndbr 6861 . . . . . . . . . . 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 676 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  E
) ( ( Q  X.c  O )  Func  T
) ( 2nd `  E
) )
4733, 34, 46funcf1 15849 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  E
) : ( ( O  Func  S )  X.  B ) --> ( Base `  T ) )
4847fovrnda 6459 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  -> 
( g ( 1st `  E ) y )  e.  ( Base `  T
) )
4948, 42eleqtrrd 2552 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  -> 
( g ( 1st `  E ) y )  e.  V )
5011, 20, 29, 43, 49elsetchom 16054 . . . . . 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 240 . . . . 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 2814 . . . 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 5879 . . . . . . 7  |-  ( z  =  <. g ,  y
>.  ->  ( M `  z )  =  ( M `  <. g ,  y >. )
)
54 df-ov 6311 . . . . . . 7  |-  ( g M y )  =  ( M `  <. g ,  y >. )
5553, 54syl6eqr 2523 . . . . . 6  |-  ( z  =  <. g ,  y
>.  ->  ( M `  z )  =  ( g M y ) )
56 fveq2 5879 . . . . . . . 8  |-  ( z  =  <. g ,  y
>.  ->  ( ( 1st `  Z ) `  z
)  =  ( ( 1st `  Z ) `
 <. g ,  y
>. ) )
57 df-ov 6311 . . . . . . . 8  |-  ( g ( 1st `  Z
) y )  =  ( ( 1st `  Z
) `  <. g ,  y >. )
5856, 57syl6eqr 2523 . . . . . . 7  |-  ( z  =  <. g ,  y
>.  ->  ( ( 1st `  Z ) `  z
)  =  ( g ( 1st `  Z
) y ) )
59 fveq2 5879 . . . . . . . 8  |-  ( z  =  <. g ,  y
>.  ->  ( ( 1st `  E ) `  z
)  =  ( ( 1st `  E ) `
 <. g ,  y
>. ) )
60 df-ov 6311 . . . . . . . 8  |-  ( g ( 1st `  E
) y )  =  ( ( 1st `  E
) `  <. g ,  y >. )
6159, 60syl6eqr 2523 . . . . . . 7  |-  ( z  =  <. g ,  y
>.  ->  ( ( 1st `  E ) `  z
)  =  ( g ( 1st `  E
) y ) )
6258, 61oveq12d 6326 . . . . . 6  |-  ( z  =  <. g ,  y
>.  ->  ( ( ( 1st `  Z ) `
 z ) ( Hom  `  T )
( ( 1st `  E
) `  z )
)  =  ( ( g ( 1st `  Z
) y ) ( Hom  `  T )
( g ( 1st `  E ) y ) ) )
6355, 62eleq12d 2543 . . . . 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 4981 . . . 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 217 . . 3  |-  ( ph  ->  A. z  e.  ( ( O  Func  S
)  X.  B ) ( M `  z
)  e.  ( ( ( 1st `  Z
) `  z )
( Hom  `  T ) ( ( 1st `  E
) `  z )
) )
66 ovex 6336 . . . . . 6  |-  ( O 
Func  S )  e.  _V
67 fvex 5889 . . . . . . 7  |-  ( Base `  C )  e.  _V
687, 67eqeltri 2545 . . . . . 6  |-  B  e. 
_V
6966, 68mpt2ex 6889 . . . . 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 2545 . . . 4  |-  M  e. 
_V
7170elixp 7547 . . 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 677 . 2  |-  ( ph  ->  M  e.  X_ z  e.  ( ( O  Func  S )  X.  B ) ( ( ( 1st `  Z ) `  z
) ( Hom  `  T
) ( ( 1st `  E ) `  z
) ) )
7317adantr 472 . . . . 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 472 . . . . 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 472 . . . . 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 472 . . . . 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 1036 . . . . . 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 6842 . . . . . 6  |-  ( z  e.  ( ( O 
Func  S )  X.  B
)  ->  ( 1st `  z )  e.  ( O  Func  S )
)
7977, 78syl 17 . . . . 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 6843 . . . . . 6  |-  ( z  e.  ( ( O 
Func  S )  X.  B
)  ->  ( 2nd `  z )  e.  B
)
8177, 80syl 17 . . . . 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 1037 . . . . . 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 6842 . . . . . 6  |-  ( w  e.  ( ( O 
Func  S )  X.  B
)  ->  ( 1st `  w )  e.  ( O  Func  S )
)
8482, 83syl 17 . . . . 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 6843 . . . . . 6  |-  ( w  e.  ( ( O 
Func  S )  X.  B
)  ->  ( 2nd `  w )  e.  B
)
8682, 85syl 17 . . . . 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 1038 . . . . . . 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 2471 . . . . . . . . . 10  |-  ( O Nat 
S )  =  ( O Nat  S )
8912, 88fuchom 15944 . . . . . . . . 9  |-  ( O Nat 
S )  =  ( Hom  `  Q )
90 eqid 2471 . . . . . . . . 9  |-  ( Hom  `  O )  =  ( Hom  `  O )
91 eqid 2471 . . . . . . . . 9  |-  ( Hom  `  ( Q  X.c  O ) )  =  ( Hom  `  ( Q  X.c  O ) )
9230, 33, 89, 90, 91, 77, 82xpchom 16143 . . . . . . . 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 2471 . . . . . . . . . 10  |-  ( Hom  `  C )  =  ( Hom  `  C )
9493, 9oppchom 15698 . . . . . . . . 9  |-  ( ( 2nd `  z ) ( Hom  `  O
) ( 2nd `  w
) )  =  ( ( 2nd `  w
) ( Hom  `  C
) ( 2nd `  z
) )
9594xpeq2i 4860 . . . . . . . 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 2521 . . . . . . 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 2551 . . . . . 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 6842 . . . . . 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 17 . . . . 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 6843 . . . . . 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 17 . . . . 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 16242 . . . 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 6849 . . . . . . . . . 10  |-  ( z  e.  ( ( O 
Func  S )  X.  B
)  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
10477, 103syl 17 . . . . . . . . 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 5883 . . . . . . . 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 6311 . . . . . . . 8  |-  ( ( 1st `  z ) ( 1st `  Z
) ( 2nd `  z
) )  =  ( ( 1st `  Z
) `  <. ( 1st `  z ) ,  ( 2nd `  z )
>. )
107105, 106syl6eqr 2523 . . . . . . 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 6849 . . . . . . . . . 10  |-  ( w  e.  ( ( O 
Func  S )  X.  B
)  ->  w  =  <. ( 1st `  w
) ,  ( 2nd `  w ) >. )
10982, 108syl 17 . . . . . . . . 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 5883 . . . . . . . 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 6311 . . . . . . . 8  |-  ( ( 1st `  w ) ( 1st `  Z
) ( 2nd `  w
) )  =  ( ( 1st `  Z
) `  <. ( 1st `  w ) ,  ( 2nd `  w )
>. )
112110, 111syl6eqr 2523 . . . . . . 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 4166 . . . . . 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 5883 . . . . . . 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 6311 . . . . . . 7  |-  ( ( 1st `  w ) ( 1st `  E
) ( 2nd `  w
) )  =  ( ( 1st `  E
) `  <. ( 1st `  w ) ,  ( 2nd `  w )
>. )
116114, 115syl6eqr 2523 . . . . . 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 6326 . . . . 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 5883 . . . . . 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 6311 . . . . . 6  |-  ( ( 1st `  w ) M ( 2nd `  w
) )  =  ( M `  <. ( 1st `  w ) ,  ( 2nd `  w
) >. )
120118, 119syl6eqr 2523 . . . . 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 6326 . . . . . . 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 6849 . . . . . . . 8  |-  ( g  e.  ( ( ( 1st `  z ) ( O Nat  S ) ( 1st `  w
) )  X.  (
( 2nd `  w
) ( Hom  `  C
) ( 2nd `  z
) ) )  -> 
g  =  <. ( 1st `  g ) ,  ( 2nd `  g
) >. )
12397, 122syl 17 . . . . . . 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 5885 . . . . . 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 6311 . . . . . 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 2523 . . . . 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 6329 . . . 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 5883 . . . . . . . 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 6311 . . . . . . . 8  |-  ( ( 1st `  z ) ( 1st `  E
) ( 2nd `  z
) )  =  ( ( 1st `  E
) `  <. ( 1st `  z ) ,  ( 2nd `  z )
>. )
130128, 129syl6eqr 2523 . . . . . . 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 4166 . . . . . 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 6326 . . . . 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 6326 . . . . . . 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 5885 . . . . . 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 6311 . . . . . 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 2523 . . . . 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 5883 . . . . . 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 6311 . . . . . 6  |-  ( ( 1st `  z ) M ( 2nd `  z
) )  =  ( M `  <. ( 1st `  z ) ,  ( 2nd `  z
) >. )
139137, 138syl6eqr 2523 . . . . 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 6329 . . . 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 2515 . . 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 2815 . 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 2471 . . 3  |-  ( ( Q  X.c  O ) Nat  T )  =  ( ( Q  X.c  O ) Nat  T )
144 eqid 2471 . . 3  |-  (comp `  T )  =  (comp `  T )
145143, 33, 91, 29, 144, 37, 44isnat2 15931 . 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 936 1  |-  ( ph  ->  M  e.  ( Z ( ( Q  X.c  O
) Nat  T ) E ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   A.wral 2756   _Vcvv 3031    u. cun 3388    C_ wss 3390   <.cop 3965   class class class wbr 4395    |-> cmpt 4454    X. cxp 4837   ran crn 4840   Rel wrel 4844    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   1stc1st 6810   2ndc2nd 6811  tpos ctpos 6990   X_cixp 7540   Basecbs 15199   Hom chom 15279  compcco 15280   Catccat 15648   Idccid 15649   Hom f chomf 15650  oppCatcoppc 15694    Func cfunc 15837    o.func ccofu 15839   Nat cnat 15924   FuncCat cfuc 15925   SetCatcsetc 16048    X.c cxpc 16131    1stF c1stf 16132    2ndF c2ndf 16133   ⟨,⟩F cprf 16134   evalF cevlf 16172  HomFchof 16211  Yoncyon 16212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-tpos 6991  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-fz 11811  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-hom 15292  df-cco 15293  df-cat 15652  df-cid 15653  df-homf 15654  df-comf 15655  df-oppc 15695  df-ssc 15793  df-resc 15794  df-subc 15795  df-func 15841  df-cofu 15843  df-nat 15926  df-fuc 15927  df-setc 16049  df-xpc 16135  df-1stf 16136  df-2ndf 16137  df-prf 16138  df-evlf 16176  df-curf 16177  df-hof 16213  df-yon 16214
This theorem is referenced by:  yonedainv  16244
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