Theorem invinv 16430

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

Step	Hyp	Ref	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 )
6	1, 2, 3, 4, 5	invsym2 16423	. . 3 |- ( ph -> `' ( X N Y ) = ( Y N X ) )
7	6	fveq1d 6193	. 2 |- ( ph -> ( `' ( X N Y ) ` ( ( X N Y ) ` F ) ) = ( ( Y N X ) ` ( ( X N Y ) ` F ) ) )
8		isoval.n	. . . 4 |- I = ( Iso ` C )
9	1, 2, 3, 4, 5, 8	invf1o 16429	. . 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 6532	. . 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 )
12	9, 10, 11	syl2anc 693	. 2 |- ( ph -> ( `' ( X N Y ) ` ( ( X N Y ) ` F ) ) = F )
13	7, 12	eqtr3d 2658	1 |- ( ph -> ( ( Y N X ) ` ( ( X N Y ) ` F ) ) = F )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   `'ccnv 5113   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   Basecbs 15857   Catccat 16325  Invcinv 16405   Iso ciso 16406

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-cat 16329  df-cid 16330  df-sect 16407  df-inv 16408  df-iso 16409

This theorem is referenced by: (None)