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Theorem invinv 15258
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 15251 . . 3  |-  ( ph  ->  `' ( X N Y )  =  ( Y N X ) )
76fveq1d 5850 . 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 15257 . . 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 6157 . . 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 659 . 2  |-  ( ph  ->  ( `' ( X N Y ) `  ( ( X N Y ) `  F
) )  =  F )
137, 12eqtr3d 2497 1  |-  ( ph  ->  ( ( Y N X ) `  (
( X N Y ) `  F ) )  =  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   `'ccnv 4987   -1-1-onto->wf1o 5569   ` cfv 5570  (class class class)co 6270   Basecbs 14716   Catccat 15153  Invcinv 15233    Iso ciso 15234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-cat 15157  df-cid 15158  df-sect 15235  df-inv 15236  df-iso 15237
This theorem is referenced by: (None)
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