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Theorem invffval 13870
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 5632 . . . . . 6  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
4 invfval.b . . . . . 6  |-  B  =  ( Base `  C
)
53, 4syl6eqr 2416 . . . . 5  |-  ( c  =  C  ->  ( Base `  c )  =  B )
6 fveq2 5632 . . . . . . . 8  |-  ( c  =  C  ->  (Sect `  c )  =  (Sect `  C ) )
7 invfval.s . . . . . . . 8  |-  S  =  (Sect `  C )
86, 7syl6eqr 2416 . . . . . . 7  |-  ( c  =  C  ->  (Sect `  c )  =  S )
98oveqd 5998 . . . . . 6  |-  ( c  =  C  ->  (
x (Sect `  c
) y )  =  ( x S y ) )
108oveqd 5998 . . . . . . 7  |-  ( c  =  C  ->  (
y (Sect `  c
) x )  =  ( y S x ) )
1110cnveqd 4960 . . . . . 6  |-  ( c  =  C  ->  `' ( y (Sect `  c ) x )  =  `' ( y S x ) )
129, 11ineq12d 3459 . . . . 5  |-  ( c  =  C  ->  (
( x (Sect `  c ) y )  i^i  `' ( y (Sect `  c )
x ) )  =  ( ( x S y )  i^i  `' ( y S x ) ) )
135, 5, 12mpt2eq123dv 6036 . . . 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 13861 . . . 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 5646 . . . . . 6  |-  ( Base `  C )  e.  _V
164, 15eqeltri 2436 . . . . 5  |-  B  e. 
_V
1716, 16mpt2ex 6325 . . . 4  |-  ( x  e.  B ,  y  e.  B  |->  ( ( x S y )  i^i  `' ( y S x ) ) )  e.  _V
1813, 14, 17fvmpt 5709 . . 3  |-  ( C  e.  Cat  ->  (Inv `  C )  =  ( x  e.  B , 
y  e.  B  |->  ( ( x S y )  i^i  `' ( y S x ) ) ) )
192, 18syl 15 . 2  |-  ( ph  ->  (Inv `  C )  =  ( x  e.  B ,  y  e.  B  |->  ( ( x S y )  i^i  `' ( y S x ) ) ) )
201, 19syl5eq 2410 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 1647    e. wcel 1715   _Vcvv 2873    i^i cin 3237   `'ccnv 4791   ` cfv 5358  (class class class)co 5981    e. cmpt2 5983   Basecbs 13356   Catccat 13776  Sectcsect 13857  Invcinv 13858
This theorem is referenced by:  invfval  13871  isoval  13877
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-inv 13861
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