Theorem isoco 16437

Description: The composition of two isomorphisms is an isomorphism. Proposition 3.14(2) of [Adamek] p. 29. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
isoco.b	|- B = ( Base ` C )
isoco.o	|- .x. = (comp ` C )
isoco.n	|- I = ( Iso ` C )
isoco.c	|- ( ph -> C e. Cat )
isoco.x	|- ( ph -> X e. B )
isoco.y	|- ( ph -> Y e. B )
isoco.z	|- ( ph -> Z e. B )
isoco.f	|- ( ph -> F e. ( X I Y ) )
isoco.g	|- ( ph -> G e. ( Y I Z ) )

Assertion

Ref	Expression
isoco	|- ( ph -> ( G ( <. X , Y >. .x. Z ) F ) e. ( X I Z ) )

Proof of Theorem isoco

Step	Hyp	Ref	Expression
1		isoco.b	. 2 |- B = ( Base ` C )
2		eqid 2622	. 2 |- (Inv ` C ) = (Inv ` C )
3		isoco.c	. 2 |- ( ph -> C e. Cat )
4		isoco.x	. 2 |- ( ph -> X e. B )
5		isoco.z	. 2 |- ( ph -> Z e. B )
6		isoco.n	. 2 |- I = ( Iso ` C )
7		isoco.y	. . 3 |- ( ph -> Y e. B )
8		isoco.f	. . 3 |- ( ph -> F e. ( X I Y ) )
9		isoco.o	. . 3 |- .x. = (comp ` C )
10		isoco.g	. . 3 |- ( ph -> G e. ( Y I Z ) )
11	1, 2, 3, 4, 7, 6, 8, 9, 5, 10	invco 16431	. 2 |- ( ph -> ( G ( <. X , Y >. .x. Z ) F ) ( X (Inv ` C ) Z ) ( ( ( X (Inv ` C ) Y ) ` F ) ( <. Z , Y >. .x. X ) ( ( Y (Inv ` C ) Z ) ` G ) ) )
12	1, 2, 3, 4, 5, 6, 11	inviso1 16426	1 |- ( ph -> ( G ( <. X , Y >. .x. Z ) F ) e. ( X I Z ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   <.cop 4183   ` cfv 5888  (class class class)co 6650   Basecbs 15857  compcco 15953   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:  cictr  16465