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Metamath Proof Explorer

Theorem oppciso

Description: An isomorphism in the opposite category. See also remark 3.9 in Adamek p. 28. (Contributed by Mario Carneiro, 3-Jan-2017)

Ref Expression
Hypotheses oppcsect.b 
|- B = ( Base ` C )
oppcsect.o 
|- O = ( oppCat ` C )
oppcsect.c 
|- ( ph -> C e. Cat )
oppcsect.x 
|- ( ph -> X e. B )
oppcsect.y 
|- ( ph -> Y e. B )
oppciso.s 
|- I = ( Iso ` C )
oppciso.t 
|- J = ( Iso ` O )
Assertion oppciso 
|- ( ph -> ( X J Y ) = ( Y I X ) )

Proof

Step Hyp Ref Expression
1 oppcsect.b 
 |-  B = ( Base ` C )
2 oppcsect.o 
 |-  O = ( oppCat ` C )
3 oppcsect.c 
 |-  ( ph -> C e. Cat )
4 oppcsect.x 
 |-  ( ph -> X e. B )
5 oppcsect.y 
 |-  ( ph -> Y e. B )
6 oppciso.s 
 |-  I = ( Iso ` C )
7 oppciso.t 
 |-  J = ( Iso ` O )
8 eqid 
 |-  ( Inv ` C ) = ( Inv ` C )
9 eqid 
 |-  ( Inv ` O ) = ( Inv ` O )
10 1 2 3 4 5 8 9 oppcinv 
 |-  ( ph -> ( X ( Inv ` O ) Y ) = ( Y ( Inv ` C ) X ) )
11 10 dmeqd 
 |-  ( ph -> dom ( X ( Inv ` O ) Y ) = dom ( Y ( Inv ` C ) X ) )
12 2 1 oppcbas 
 |-  B = ( Base ` O )
13 2 oppccat 
 |-  ( C e. Cat -> O e. Cat )
14 3 13 syl 
 |-  ( ph -> O e. Cat )
15 12 9 14 4 5 7 isoval 
 |-  ( ph -> ( X J Y ) = dom ( X ( Inv ` O ) Y ) )
16 1 8 3 5 4 6 isoval 
 |-  ( ph -> ( Y I X ) = dom ( Y ( Inv ` C ) X ) )
17 11 15 16 3eqtr4d 
 |-  ( ph -> ( X J Y ) = ( Y I X ) )