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Theorem invffval 13988
Description: Value of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
invfval.b  |-  B  =  ( Base `  C
)
invfval.n  |-  N  =  (Inv `  C )
invfval.c  |-  ( ph  ->  C  e.  Cat )
invfval.x  |-  ( ph  ->  X  e.  B )
invfval.y  |-  ( ph  ->  Y  e.  B )
invfval.s  |-  S  =  (Sect `  C )
Assertion
Ref Expression
invffval  |-  ( ph  ->  N  =  ( x  e.  B ,  y  e.  B  |->  ( ( x S y )  i^i  `' ( y S x ) ) ) )
Distinct variable groups:    x, y, B    ph, x, y    x, X, y    x, Y, y   
x, C, y    x, S, y
Allowed substitution hints:    N( x, y)

Proof of Theorem invffval
Dummy variable  c is distinct from all other variables.
StepHypRef Expression
1 invfval.n . 2  |-  N  =  (Inv `  C )
2 invfval.c . . 3  |-  ( ph  ->  C  e.  Cat )
3 fveq2 5731 . . . . . 6  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
4 invfval.b . . . . . 6  |-  B  =  ( Base `  C
)
53, 4syl6eqr 2488 . . . . 5  |-  ( c  =  C  ->  ( Base `  c )  =  B )
6 fveq2 5731 . . . . . . . 8  |-  ( c  =  C  ->  (Sect `  c )  =  (Sect `  C ) )
7 invfval.s . . . . . . . 8  |-  S  =  (Sect `  C )
86, 7syl6eqr 2488 . . . . . . 7  |-  ( c  =  C  ->  (Sect `  c )  =  S )
98oveqd 6101 . . . . . 6  |-  ( c  =  C  ->  (
x (Sect `  c
) y )  =  ( x S y ) )
108oveqd 6101 . . . . . . 7  |-  ( c  =  C  ->  (
y (Sect `  c
) x )  =  ( y S x ) )
1110cnveqd 5051 . . . . . 6  |-  ( c  =  C  ->  `' ( y (Sect `  c ) x )  =  `' ( y S x ) )
129, 11ineq12d 3545 . . . . 5  |-  ( c  =  C  ->  (
( x (Sect `  c ) y )  i^i  `' ( y (Sect `  c )
x ) )  =  ( ( x S y )  i^i  `' ( y S x ) ) )
135, 5, 12mpt2eq123dv 6139 . . . 4  |-  ( c  =  C  ->  (
x  e.  ( Base `  c ) ,  y  e.  ( Base `  c
)  |->  ( ( x (Sect `  c )
y )  i^i  `' ( y (Sect `  c ) x ) ) )  =  ( x  e.  B , 
y  e.  B  |->  ( ( x S y )  i^i  `' ( y S x ) ) ) )
14 df-inv 13979 . . . 4  |- Inv  =  ( c  e.  Cat  |->  ( x  e.  ( Base `  c ) ,  y  e.  ( Base `  c
)  |->  ( ( x (Sect `  c )
y )  i^i  `' ( y (Sect `  c ) x ) ) ) )
15 fvex 5745 . . . . . 6  |-  ( Base `  C )  e.  _V
164, 15eqeltri 2508 . . . . 5  |-  B  e. 
_V
1716, 16mpt2ex 6428 . . . 4  |-  ( x  e.  B ,  y  e.  B  |->  ( ( x S y )  i^i  `' ( y S x ) ) )  e.  _V
1813, 14, 17fvmpt 5809 . . 3  |-  ( C  e.  Cat  ->  (Inv `  C )  =  ( x  e.  B , 
y  e.  B  |->  ( ( x S y )  i^i  `' ( y S x ) ) ) )
192, 18syl 16 . 2  |-  ( ph  ->  (Inv `  C )  =  ( x  e.  B ,  y  e.  B  |->  ( ( x S y )  i^i  `' ( y S x ) ) ) )
201, 19syl5eq 2482 1  |-  ( ph  ->  N  =  ( x  e.  B ,  y  e.  B  |->  ( ( x S y )  i^i  `' ( y S x ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   _Vcvv 2958    i^i cin 3321   `'ccnv 4880   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086   Basecbs 13474   Catccat 13894  Sectcsect 13975  Invcinv 13976
This theorem is referenced by:  invfval  13989  isoval  13995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-inv 13979
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