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Theorem invfval 13986
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 13985 . 2  |-  ( ph  ->  N  =  ( x  e.  B ,  y  e.  B  |->  ( ( x S y )  i^i  `' ( y S x ) ) ) )
7 simprl 734 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  x  =  X )
8 simprr 735 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
y  =  Y )
97, 8oveq12d 6101 . . 3  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( x S y )  =  ( X S Y ) )
108, 7oveq12d 6101 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( y S x )  =  ( Y S X ) )
1110cnveqd 5050 . . 3  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  `' ( y S x )  =  `' ( Y S X ) )
129, 11ineq12d 3545 . 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 6108 . . . 4  |-  ( X S Y )  e. 
_V
1514inex1 4346 . . 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 6203 1  |-  ( ph  ->  ( X N Y )  =  ( ( X S Y )  i^i  `' ( Y S X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958    i^i cin 3321   `'ccnv 4879   ` cfv 5456  (class class class)co 6083   Basecbs 13471   Catccat 13891  Sectcsect 13972  Invcinv 13973
This theorem is referenced by:  isinv  13987  invss  13988  oppcinv  14003
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-inv 13976
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