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Theorem invffval 13942
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 5691 . . . . . 6  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
4 invfval.b . . . . . 6  |-  B  =  ( Base `  C
)
53, 4syl6eqr 2458 . . . . 5  |-  ( c  =  C  ->  ( Base `  c )  =  B )
6 fveq2 5691 . . . . . . . 8  |-  ( c  =  C  ->  (Sect `  c )  =  (Sect `  C ) )
7 invfval.s . . . . . . . 8  |-  S  =  (Sect `  C )
86, 7syl6eqr 2458 . . . . . . 7  |-  ( c  =  C  ->  (Sect `  c )  =  S )
98oveqd 6061 . . . . . 6  |-  ( c  =  C  ->  (
x (Sect `  c
) y )  =  ( x S y ) )
108oveqd 6061 . . . . . . 7  |-  ( c  =  C  ->  (
y (Sect `  c
) x )  =  ( y S x ) )
1110cnveqd 5011 . . . . . 6  |-  ( c  =  C  ->  `' ( y (Sect `  c ) x )  =  `' ( y S x ) )
129, 11ineq12d 3507 . . . . 5  |-  ( c  =  C  ->  (
( x (Sect `  c ) y )  i^i  `' ( y (Sect `  c )
x ) )  =  ( ( x S y )  i^i  `' ( y S x ) ) )
135, 5, 12mpt2eq123dv 6099 . . . 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 13933 . . . 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 5705 . . . . . 6  |-  ( Base `  C )  e.  _V
164, 15eqeltri 2478 . . . . 5  |-  B  e. 
_V
1716, 16mpt2ex 6388 . . . 4  |-  ( x  e.  B ,  y  e.  B  |->  ( ( x S y )  i^i  `' ( y S x ) ) )  e.  _V
1813, 14, 17fvmpt 5769 . . 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 2452 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 1649    e. wcel 1721   _Vcvv 2920    i^i cin 3283   `'ccnv 4840   ` cfv 5417  (class class class)co 6044    e. cmpt2 6046   Basecbs 13428   Catccat 13848  Sectcsect 13929  Invcinv 13930
This theorem is referenced by:  invfval  13943  isoval  13949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-inv 13933
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