Metamath Proof Explorer


Theorem initoeu1w

Description: Initial objects are essentially unique (weak form), i.e. if A and B are initial objects, then A and B are isomorphic. Proposition 7.3 (1) of Adamek p. 102. (Contributed by AV, 6-Apr-2020)

Ref Expression
Hypotheses initoeu1.c
|- ( ph -> C e. Cat )
initoeu1.a
|- ( ph -> A e. ( InitO ` C ) )
initoeu1.b
|- ( ph -> B e. ( InitO ` C ) )
Assertion initoeu1w
|- ( ph -> A ( ~=c ` C ) B )

Proof

Step Hyp Ref Expression
1 initoeu1.c
 |-  ( ph -> C e. Cat )
2 initoeu1.a
 |-  ( ph -> A e. ( InitO ` C ) )
3 initoeu1.b
 |-  ( ph -> B e. ( InitO ` C ) )
4 1 2 3 initoeu1
 |-  ( ph -> E! f f e. ( A ( Iso ` C ) B ) )
5 euex
 |-  ( E! f f e. ( A ( Iso ` C ) B ) -> E. f f e. ( A ( Iso ` C ) B ) )
6 4 5 syl
 |-  ( ph -> E. f f e. ( A ( Iso ` C ) B ) )
7 eqid
 |-  ( Iso ` C ) = ( Iso ` C )
8 eqid
 |-  ( Base ` C ) = ( Base ` C )
9 initoo
 |-  ( C e. Cat -> ( A e. ( InitO ` C ) -> A e. ( Base ` C ) ) )
10 1 2 9 sylc
 |-  ( ph -> A e. ( Base ` C ) )
11 initoo
 |-  ( C e. Cat -> ( B e. ( InitO ` C ) -> B e. ( Base ` C ) ) )
12 1 3 11 sylc
 |-  ( ph -> B e. ( Base ` C ) )
13 7 8 1 10 12 cic
 |-  ( ph -> ( A ( ~=c ` C ) B <-> E. f f e. ( A ( Iso ` C ) B ) ) )
14 6 13 mpbird
 |-  ( ph -> A ( ~=c ` C ) B )