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Theorem invffval 14819
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 5802 . . . . . 6  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
4 invfval.b . . . . . 6  |-  B  =  ( Base `  C
)
53, 4syl6eqr 2513 . . . . 5  |-  ( c  =  C  ->  ( Base `  c )  =  B )
6 fveq2 5802 . . . . . . . 8  |-  ( c  =  C  ->  (Sect `  c )  =  (Sect `  C ) )
7 invfval.s . . . . . . . 8  |-  S  =  (Sect `  C )
86, 7syl6eqr 2513 . . . . . . 7  |-  ( c  =  C  ->  (Sect `  c )  =  S )
98oveqd 6220 . . . . . 6  |-  ( c  =  C  ->  (
x (Sect `  c
) y )  =  ( x S y ) )
108oveqd 6220 . . . . . . 7  |-  ( c  =  C  ->  (
y (Sect `  c
) x )  =  ( y S x ) )
1110cnveqd 5126 . . . . . 6  |-  ( c  =  C  ->  `' ( y (Sect `  c ) x )  =  `' ( y S x ) )
129, 11ineq12d 3664 . . . . 5  |-  ( c  =  C  ->  (
( x (Sect `  c ) y )  i^i  `' ( y (Sect `  c )
x ) )  =  ( ( x S y )  i^i  `' ( y S x ) ) )
135, 5, 12mpt2eq123dv 6260 . . . 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 14810 . . . 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 5812 . . . . . 6  |-  ( Base `  C )  e.  _V
164, 15eqeltri 2538 . . . . 5  |-  B  e. 
_V
1716, 16mpt2ex 6763 . . . 4  |-  ( x  e.  B ,  y  e.  B  |->  ( ( x S y )  i^i  `' ( y S x ) ) )  e.  _V
1813, 14, 17fvmpt 5886 . . 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 2507 1  |-  ( ph  ->  N  =  ( x  e.  B ,  y  e.  B  |->  ( ( x S y )  i^i  `' ( y S x ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   _Vcvv 3078    i^i cin 3438   `'ccnv 4950   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205   Basecbs 14296   Catccat 14725  Sectcsect 14806  Invcinv 14807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-inv 14810
This theorem is referenced by:  invfval  14820  isoval  14826
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