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Theorem invfval 13677
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 13676 . 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 5892 . . 3  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( x S y )  =  ( X S Y ) )
108, 7oveq12d 5892 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( y S x )  =  ( Y S X ) )
1110cnveqd 4873 . . 3  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  `' ( y S x )  =  `' ( Y S X ) )
129, 11ineq12d 3384 . 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 5899 . . . 4  |-  ( X S Y )  e. 
_V
1514inex1 4171 . . 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 5991 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 1632    e. wcel 1696   _Vcvv 2801    i^i cin 3164   `'ccnv 4704   ` cfv 5271  (class class class)co 5874   Basecbs 13164   Catccat 13582  Sectcsect 13663  Invcinv 13664
This theorem is referenced by:  isinv  13678  invss  13679  oppcinv  13694
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-inv 13667
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