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Metamath Proof Explorer

Theorem isofval2

Description: Function value of the function returning the isomorphisms of a category. (Contributed by Zhi Wang, 27-Oct-2025)

Ref Expression
Hypotheses isofval2.b 
|- B = ( Base ` C )
isofval2.n 
|- N = ( Inv ` C )
isofval2.c 
|- ( ph -> C e. Cat )
isofval2.i 
|- I = ( Iso ` C )
Assertion isofval2 
|- ( ph -> I = ( x e. B , y e. B |-> dom ( x N y ) ) )

Proof

Step Hyp Ref Expression
1 isofval2.b 
 |-  B = ( Base ` C )
2 isofval2.n 
 |-  N = ( Inv ` C )
3 isofval2.c 
 |-  ( ph -> C e. Cat )
4 isofval2.i 
 |-  I = ( Iso ` C )
5 isofn 
 |-  ( C e. Cat -> ( Iso ` C ) Fn ( ( Base ` C ) X. ( Base ` C ) ) )
6 4 fneq1i 
 |-  ( I Fn ( B X. B ) <-> ( Iso ` C ) Fn ( B X. B ) )
7 1 1 xpeq12i 
 |-  ( B X. B ) = ( ( Base ` C ) X. ( Base ` C ) )
8 7 fneq2i 
 |-  ( ( Iso ` C ) Fn ( B X. B ) <-> ( Iso ` C ) Fn ( ( Base ` C ) X. ( Base ` C ) ) )
9 6 8 bitri 
 |-  ( I Fn ( B X. B ) <-> ( Iso ` C ) Fn ( ( Base ` C ) X. ( Base ` C ) ) )
10 5 9 sylibr 
 |-  ( C e. Cat -> I Fn ( B X. B ) )
11 fnov 
 |-  ( I Fn ( B X. B ) <-> I = ( x e. B , y e. B |-> ( x I y ) ) )
12 10 11 sylib 
 |-  ( C e. Cat -> I = ( x e. B , y e. B |-> ( x I y ) ) )
13 3 12 syl 
 |-  ( ph -> I = ( x e. B , y e. B |-> ( x I y ) ) )
14 3 3ad2ant1 
 |-  ( ( ph /\ x e. B /\ y e. B ) -> C e. Cat )
15 simp2 
 |-  ( ( ph /\ x e. B /\ y e. B ) -> x e. B )
16 simp3 
 |-  ( ( ph /\ x e. B /\ y e. B ) -> y e. B )
17 1 2 14 15 16 4 isoval 
 |-  ( ( ph /\ x e. B /\ y e. B ) -> ( x I y ) = dom ( x N y ) )
18 17 mpoeq3dva 
 |-  ( ph -> ( x e. B , y e. B |-> ( x I y ) ) = ( x e. B , y e. B |-> dom ( x N y ) ) )
19 13 18 eqtrd 
 |-  ( ph -> I = ( x e. B , y e. B |-> dom ( x N y ) ) )