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Theorem invfval 15264
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
invfval  |-  ( ph  ->  ( X N Y )  =  ( ( X S Y )  i^i  `' ( Y S X ) ) )

Proof of Theorem invfval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 invfval.b . . 3  |-  B  =  ( Base `  C
)
2 invfval.n . . 3  |-  N  =  (Inv `  C )
3 invfval.c . . 3  |-  ( ph  ->  C  e.  Cat )
4 invfval.x . . 3  |-  ( ph  ->  X  e.  B )
5 invfval.s . . 3  |-  S  =  (Sect `  C )
61, 2, 3, 4, 4, 5invffval 15263 . 2  |-  ( ph  ->  N  =  ( x  e.  B ,  y  e.  B  |->  ( ( x S y )  i^i  `' ( y S x ) ) ) )
7 simprl 756 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  x  =  X )
8 simprr 758 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
y  =  Y )
97, 8oveq12d 6252 . . 3  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( x S y )  =  ( X S Y ) )
108, 7oveq12d 6252 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( y S x )  =  ( Y S X ) )
1110cnveqd 5120 . . 3  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  `' ( y S x )  =  `' ( Y S X ) )
129, 11ineq12d 3641 . 2  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( ( x S y )  i^i  `' ( y S x ) )  =  ( ( X S Y )  i^i  `' ( Y S X ) ) )
13 invfval.y . 2  |-  ( ph  ->  Y  e.  B )
14 ovex 6262 . . . 4  |-  ( X S Y )  e. 
_V
1514inex1 4534 . . 3  |-  ( ( X S Y )  i^i  `' ( Y S X ) )  e.  _V
1615a1i 11 . 2  |-  ( ph  ->  ( ( X S Y )  i^i  `' ( Y S X ) )  e.  _V )
176, 12, 4, 13, 16ovmpt2d 6367 1  |-  ( ph  ->  ( X N Y )  =  ( ( X S Y )  i^i  `' ( Y S X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   _Vcvv 3058    i^i cin 3412   `'ccnv 4941   ` cfv 5525  (class class class)co 6234   Basecbs 14733   Catccat 15170  Sectcsect 15249  Invcinv 15250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-1st 6738  df-2nd 6739  df-inv 15253
This theorem is referenced by:  isinv  15265  invss  15266  dfiso2  15277  oppcinv  15285
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