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Theorem invinv 13688
Description: The inverse of the inverse of an isomorphism is itself. Proposition 3.14(1) of [Adamek] p. 29. (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 )
isoval.n  |-  I  =  (  Iso  `  C
)
invinv.f  |-  ( ph  ->  F  e.  ( X I Y ) )
Assertion
Ref Expression
invinv  |-  ( ph  ->  ( ( Y N X ) `  (
( X N Y ) `  F ) )  =  F )

Proof of Theorem invinv
StepHypRef Expression
1 invfval.b . . . 4  |-  B  =  ( Base `  C
)
2 invfval.n . . . 4  |-  N  =  (Inv `  C )
3 invfval.c . . . 4  |-  ( ph  ->  C  e.  Cat )
4 invfval.x . . . 4  |-  ( ph  ->  X  e.  B )
5 invfval.y . . . 4  |-  ( ph  ->  Y  e.  B )
61, 2, 3, 4, 5invsym2 13681 . . 3  |-  ( ph  ->  `' ( X N Y )  =  ( Y N X ) )
76fveq1d 5543 . 2  |-  ( ph  ->  ( `' ( X N Y ) `  ( ( X N Y ) `  F
) )  =  ( ( Y N X ) `  ( ( X N Y ) `
 F ) ) )
8 isoval.n . . . 4  |-  I  =  (  Iso  `  C
)
91, 2, 3, 4, 5, 8invf1o 13687 . . 3  |-  ( ph  ->  ( X N Y ) : ( X I Y ) -1-1-onto-> ( Y I X ) )
10 invinv.f . . 3  |-  ( ph  ->  F  e.  ( X I Y ) )
11 f1ocnvfv1 5808 . . 3  |-  ( ( ( X N Y ) : ( X I Y ) -1-1-onto-> ( Y I X )  /\  F  e.  ( X I Y ) )  -> 
( `' ( X N Y ) `  ( ( X N Y ) `  F
) )  =  F )
129, 10, 11syl2anc 642 . 2  |-  ( ph  ->  ( `' ( X N Y ) `  ( ( X N Y ) `  F
) )  =  F )
137, 12eqtr3d 2330 1  |-  ( ph  ->  ( ( Y N X ) `  (
( X N Y ) `  F ) )  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   `'ccnv 4704   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   Basecbs 13164   Catccat 13582  Invcinv 13664    Iso ciso 13665
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-rmo 2564  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-riota 6320  df-cat 13586  df-cid 13587  df-sect 13666  df-inv 13667  df-iso 13668
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