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Theorem invfval 13661
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 13660 . 2  |-  ( ph  ->  N  =  ( x  e.  B ,  y  e.  B  |->  ( ( x S y )  i^i  `' ( y S x ) ) ) )
7 simprl 732 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  x  =  X )
8 simprr 733 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
y  =  Y )
97, 8oveq12d 5876 . . 3  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( x S y )  =  ( X S Y ) )
108, 7oveq12d 5876 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( y S x )  =  ( Y S X ) )
1110cnveqd 4857 . . 3  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  `' ( y S x )  =  `' ( Y S X ) )
129, 11ineq12d 3371 . 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 5883 . . . 4  |-  ( X S Y )  e. 
_V
1514inex1 4155 . . 3  |-  ( ( X S Y )  i^i  `' ( Y S X ) )  e.  _V
1615a1i 10 . 2  |-  ( ph  ->  ( ( X S Y )  i^i  `' ( Y S X ) )  e.  _V )
176, 12, 4, 13, 16ovmpt2d 5975 1  |-  ( ph  ->  ( X N Y )  =  ( ( X S Y )  i^i  `' ( Y S X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151   `'ccnv 4688   ` cfv 5255  (class class class)co 5858   Basecbs 13148   Catccat 13566  Sectcsect 13647  Invcinv 13648
This theorem is referenced by:  isinv  13662  invss  13663  oppcinv  13678
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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