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Theorem invfval 13939
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 13938 . 2  |-  ( ph  ->  N  =  ( x  e.  B ,  y  e.  B  |->  ( ( x S y )  i^i  `' ( y S x ) ) ) )
7 simprl 733 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  x  =  X )
8 simprr 734 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
y  =  Y )
97, 8oveq12d 6058 . . 3  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( x S y )  =  ( X S Y ) )
108, 7oveq12d 6058 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( y S x )  =  ( Y S X ) )
1110cnveqd 5007 . . 3  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  `' ( y S x )  =  `' ( Y S X ) )
129, 11ineq12d 3503 . 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 6065 . . . 4  |-  ( X S Y )  e. 
_V
1514inex1 4304 . . 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 6160 1  |-  ( ph  ->  ( X N Y )  =  ( ( X S Y )  i^i  `' ( Y S X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916    i^i cin 3279   `'ccnv 4836   ` cfv 5413  (class class class)co 6040   Basecbs 13424   Catccat 13844  Sectcsect 13925  Invcinv 13926
This theorem is referenced by:  isinv  13940  invss  13941  oppcinv  13956
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-inv 13929
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