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Theorem yonedalem3 15423
Description: Lemma for yoneda 15426. (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 6309 . . . . . 6  |-  ( ( ( 1st `  Y
) `  x )
( O Nat  S ) f )  e.  _V
32mptex 6128 . . . . 5  |-  ( a  e.  ( ( ( 1st `  Y ) `
 x ) ( O Nat  S ) f )  |->  ( ( a `
 x ) `  (  .1.  `  x )
) )  e.  _V
41, 3fnmpt2i 6854 . . . 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 465 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  ->  C  e.  Cat )
19 yoneda.w . . . . . . . . 9  |-  ( ph  ->  V  e.  W )
2019adantr 465 . . . . . . . 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 465 . . . . . . . 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 465 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  -> 
( ran  ( Hom f  `  Q
)  u.  U ) 
C_  V )
25 simprl 756 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  -> 
g  e.  ( O 
Func  S ) )
26 simprr 757 . . . . . . . 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 15417 . . . . . . 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 463 . . . . . 6  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  -> 
( g M y ) : ( g ( 1st `  Z
) y ) --> ( g ( 1st `  E
) y ) )
29 eqid 2443 . . . . . . 7  |-  ( Hom  `  T )  =  ( Hom  `  T )
30 eqid 2443 . . . . . . . . . . 11  |-  ( Q  X.c  O )  =  ( Q  X.c  O )
3112fucbas 15203 . . . . . . . . . . 11  |-  ( O 
Func  S )  =  (
Base `  Q )
329, 7oppcbas 14990 . . . . . . . . . . 11  |-  B  =  ( Base `  O
)
3330, 31, 32xpcbas 15321 . . . . . . . . . 10  |-  ( ( O  Func  S )  X.  B )  =  (
Base `  ( Q  X.c  O ) )
34 eqid 2443 . . . . . . . . . 10  |-  ( Base `  T )  =  (
Base `  T )
35 relfunc 15105 . . . . . . . . . . 11  |-  Rel  (
( Q  X.c  O ) 
Func  T )
366, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 23yonedalem1 15415 . . . . . . . . . . . 12  |-  ( ph  ->  ( Z  e.  ( ( Q  X.c  O ) 
Func  T )  /\  E  e.  ( ( Q  X.c  O
)  Func  T )
) )
3736simpld 459 . . . . . . . . . . 11  |-  ( ph  ->  Z  e.  ( ( Q  X.c  O )  Func  T
) )
38 1st2ndbr 6834 . . . . . . . . . . 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 663 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  Z
) ( ( Q  X.c  O )  Func  T
) ( 2nd `  Z
) )
4033, 34, 39funcf1 15109 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  Z
) : ( ( O  Func  S )  X.  B ) --> ( Base `  T ) )
4140fovrnda 6431 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  -> 
( g ( 1st `  Z ) y )  e.  ( Base `  T
) )
4211, 20setcbas 15279 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  ->  V  =  ( Base `  T ) )
4341, 42eleqtrrd 2534 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  -> 
( g ( 1st `  Z ) y )  e.  V )
4436simprd 463 . . . . . . . . . . 11  |-  ( ph  ->  E  e.  ( ( Q  X.c  O )  Func  T
) )
45 1st2ndbr 6834 . . . . . . . . . . 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 663 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  E
) ( ( Q  X.c  O )  Func  T
) ( 2nd `  E
) )
4733, 34, 46funcf1 15109 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  E
) : ( ( O  Func  S )  X.  B ) --> ( Base `  T ) )
4847fovrnda 6431 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  -> 
( g ( 1st `  E ) y )  e.  ( Base `  T
) )
4948, 42eleqtrrd 2534 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  ( O  Func  S
)  /\  y  e.  B ) )  -> 
( g ( 1st `  E ) y )  e.  V )
5011, 20, 29, 43, 49elsetchom 15282 . . . . . 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 2864 . . . 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 5856 . . . . . . 7  |-  ( z  =  <. g ,  y
>.  ->  ( M `  z )  =  ( M `  <. g ,  y >. )
)
54 df-ov 6284 . . . . . . 7  |-  ( g M y )  =  ( M `  <. g ,  y >. )
5553, 54syl6eqr 2502 . . . . . 6  |-  ( z  =  <. g ,  y
>.  ->  ( M `  z )  =  ( g M y ) )
56 fveq2 5856 . . . . . . . 8  |-  ( z  =  <. g ,  y
>.  ->  ( ( 1st `  Z ) `  z
)  =  ( ( 1st `  Z ) `
 <. g ,  y
>. ) )
57 df-ov 6284 . . . . . . . 8  |-  ( g ( 1st `  Z
) y )  =  ( ( 1st `  Z
) `  <. g ,  y >. )
5856, 57syl6eqr 2502 . . . . . . 7  |-  ( z  =  <. g ,  y
>.  ->  ( ( 1st `  Z ) `  z
)  =  ( g ( 1st `  Z
) y ) )
59 fveq2 5856 . . . . . . . 8  |-  ( z  =  <. g ,  y
>.  ->  ( ( 1st `  E ) `  z
)  =  ( ( 1st `  E ) `
 <. g ,  y
>. ) )
60 df-ov 6284 . . . . . . . 8  |-  ( g ( 1st `  E
) y )  =  ( ( 1st `  E
) `  <. g ,  y >. )
6159, 60syl6eqr 2502 . . . . . . 7  |-  ( z  =  <. g ,  y
>.  ->  ( ( 1st `  E ) `  z
)  =  ( g ( 1st `  E
) y ) )
6258, 61oveq12d 6299 . . . . . 6  |-  ( z  =  <. g ,  y
>.  ->  ( ( ( 1st `  Z ) `
 z ) ( Hom  `  T )
( ( 1st `  E
) `  z )
)  =  ( ( g ( 1st `  Z
) y ) ( Hom  `  T )
( g ( 1st `  E ) y ) ) )
6355, 62eleq12d 2525 . . . . 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 5134 . . . 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 6309 . . . . . 6  |-  ( O 
Func  S )  e.  _V
67 fvex 5866 . . . . . . 7  |-  ( Base `  C )  e.  _V
687, 67eqeltri 2527 . . . . . 6  |-  B  e. 
_V
6966, 68mpt2ex 6862 . . . . 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 2527 . . . 4  |-  M  e. 
_V
7170elixp 7478 . . 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 664 . 2  |-  ( ph  ->  M  e.  X_ z  e.  ( ( O  Func  S )  X.  B ) ( ( ( 1st `  Z ) `  z
) ( Hom  `  T
) ( ( 1st `  E ) `  z
) ) )
7317adantr 465 . . . . 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 465 . . . . 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 465 . . . . 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 465 . . . . 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 1003 . . . . . 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 6815 . . . . . 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 6816 . . . . . 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 1004 . . . . . 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 6815 . . . . . 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 6816 . . . . . 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 1005 . . . . . . 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 2443 . . . . . . . . . 10  |-  ( O Nat 
S )  =  ( O Nat  S )
8912, 88fuchom 15204 . . . . . . . . 9  |-  ( O Nat 
S )  =  ( Hom  `  Q )
90 eqid 2443 . . . . . . . . 9  |-  ( Hom  `  O )  =  ( Hom  `  O )
91 eqid 2443 . . . . . . . . 9  |-  ( Hom  `  ( Q  X.c  O ) )  =  ( Hom  `  ( Q  X.c  O ) )
9230, 33, 89, 90, 91, 77, 82xpchom 15323 . . . . . . . 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 2443 . . . . . . . . . 10  |-  ( Hom  `  C )  =  ( Hom  `  C )
9493, 9oppchom 14987 . . . . . . . . 9  |-  ( ( 2nd `  z ) ( Hom  `  O
) ( 2nd `  w
) )  =  ( ( 2nd `  w
) ( Hom  `  C
) ( 2nd `  z
) )
9594xpeq2i 5010 . . . . . . . 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 2500 . . . . . . 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 2533 . . . . . 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 6815 . . . . . 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 6816 . . . . . 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 15422 . . . 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 6822 . . . . . . . . . 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 5860 . . . . . . . 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 6284 . . . . . . . 8  |-  ( ( 1st `  z ) ( 1st `  Z
) ( 2nd `  z
) )  =  ( ( 1st `  Z
) `  <. ( 1st `  z ) ,  ( 2nd `  z )
>. )
107105, 106syl6eqr 2502 . . . . . . 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 6822 . . . . . . . . . 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 5860 . . . . . . . 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 6284 . . . . . . . 8  |-  ( ( 1st `  w ) ( 1st `  Z
) ( 2nd `  w
) )  =  ( ( 1st `  Z
) `  <. ( 1st `  w ) ,  ( 2nd `  w )
>. )
112110, 111syl6eqr 2502 . . . . . . 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 4210 . . . . . 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 5860 . . . . . . 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 6284 . . . . . . 7  |-  ( ( 1st `  w ) ( 1st `  E
) ( 2nd `  w
) )  =  ( ( 1st `  E
) `  <. ( 1st `  w ) ,  ( 2nd `  w )
>. )
116114, 115syl6eqr 2502 . . . . . 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 6299 . . . . 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 5860 . . . . . 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 6284 . . . . . 6  |-  ( ( 1st `  w ) M ( 2nd `  w
) )  =  ( M `  <. ( 1st `  w ) ,  ( 2nd `  w
) >. )
120118, 119syl6eqr 2502 . . . . 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 6299 . . . . . . 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 6822 . . . . . . . 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 5862 . . . . . 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 6284 . . . . . 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 2502 . . . . 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 6302 . . . 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 5860 . . . . . . . 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 6284 . . . . . . . 8  |-  ( ( 1st `  z ) ( 1st `  E
) ( 2nd `  z
) )  =  ( ( 1st `  E
) `  <. ( 1st `  z ) ,  ( 2nd `  z )
>. )
130128, 129syl6eqr 2502 . . . . . . 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 4210 . . . . . 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 6299 . . . . 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 6299 . . . . . . 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 5862 . . . . . 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 6284 . . . . . 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 2502 . . . . 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 5860 . . . . . 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 6284 . . . . . 6  |-  ( ( 1st `  z ) M ( 2nd `  z
) )  =  ( M `  <. ( 1st `  z ) ,  ( 2nd `  z
) >. )
139137, 138syl6eqr 2502 . . . . 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 6302 . . . 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 2494 . . 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 2865 . 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 2443 . . 3  |-  ( ( Q  X.c  O ) Nat  T )  =  ( ( Q  X.c  O ) Nat  T )
144 eqid 2443 . . 3  |-  (comp `  T )  =  (comp `  T )
145143, 33, 91, 29, 144, 37, 44isnat2 15191 . 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 922 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 974    = wceq 1383    e. wcel 1804   A.wral 2793   _Vcvv 3095    u. cun 3459    C_ wss 3461   <.cop 4020   class class class wbr 4437    |-> cmpt 4495    X. cxp 4987   ran crn 4990   Rel wrel 4994    Fn wfn 5573   -->wf 5574   ` cfv 5578  (class class class)co 6281    |-> cmpt2 6283   1stc1st 6783   2ndc2nd 6784  tpos ctpos 6956   X_cixp 7471   Basecbs 14509   Hom chom 14585  compcco 14586   Catccat 14938   Idccid 14939   Hom f chomf 14940  oppCatcoppc 14983    Func cfunc 15097    o.func ccofu 15099   Nat cnat 15184   FuncCat cfuc 15185   SetCatcsetc 15276    X.c cxpc 15311    1stF c1stf 15312    2ndF c2ndf 15313   ⟨,⟩F cprf 15314   evalF cevlf 15352  HomFchof 15391  Yoncyon 15392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-tpos 6957  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-dec 10985  df-uz 11091  df-fz 11682  df-struct 14511  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-hom 14598  df-cco 14599  df-cat 14942  df-cid 14943  df-homf 14944  df-comf 14945  df-oppc 14984  df-ssc 15056  df-resc 15057  df-subc 15058  df-func 15101  df-cofu 15103  df-nat 15186  df-fuc 15187  df-setc 15277  df-xpc 15315  df-1stf 15316  df-2ndf 15317  df-prf 15318  df-evlf 15356  df-curf 15357  df-hof 15393  df-yon 15394
This theorem is referenced by:  yonedainv  15424
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