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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 )