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Theorem invffval 13660
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 5525 . . . . . 6  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
4 invfval.b . . . . . 6  |-  B  =  ( Base `  C
)
53, 4syl6eqr 2333 . . . . 5  |-  ( c  =  C  ->  ( Base `  c )  =  B )
6 fveq2 5525 . . . . . . . 8  |-  ( c  =  C  ->  (Sect `  c )  =  (Sect `  C ) )
7 invfval.s . . . . . . . 8  |-  S  =  (Sect `  C )
86, 7syl6eqr 2333 . . . . . . 7  |-  ( c  =  C  ->  (Sect `  c )  =  S )
98oveqd 5875 . . . . . 6  |-  ( c  =  C  ->  (
x (Sect `  c
) y )  =  ( x S y ) )
108oveqd 5875 . . . . . . 7  |-  ( c  =  C  ->  (
y (Sect `  c
) x )  =  ( y S x ) )
1110cnveqd 4857 . . . . . 6  |-  ( c  =  C  ->  `' ( y (Sect `  c ) x )  =  `' ( y S x ) )
129, 11ineq12d 3371 . . . . 5  |-  ( c  =  C  ->  (
( x (Sect `  c ) y )  i^i  `' ( y (Sect `  c )
x ) )  =  ( ( x S y )  i^i  `' ( y S x ) ) )
135, 5, 12mpt2eq123dv 5910 . . . 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 13651 . . . 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 5539 . . . . . 6  |-  ( Base `  C )  e.  _V
164, 15eqeltri 2353 . . . . 5  |-  B  e. 
_V
1716, 16mpt2ex 6198 . . . 4  |-  ( x  e.  B ,  y  e.  B  |->  ( ( x S y )  i^i  `' ( y S x ) ) )  e.  _V
1813, 14, 17fvmpt 5602 . . 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 2327 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 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151   `'ccnv 4688   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   Basecbs 13148   Catccat 13566  Sectcsect 13647  Invcinv 13648
This theorem is referenced by:  invfval  13661  isoval  13667
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-inv 13651
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