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Metamath Proof Explorer

Theorem wrdpmcl

Description: Closure of a word with permuted symbols. (Contributed by Thierry Arnoux, 27-May-2025)

Ref Expression
Hypotheses wrdpmcl.1 
|- J = ( 0 ..^ ( # ` W ) )
wrdpmcl.2 
|- ( ph -> E : J -1-1-onto-> J )
wrdpmcl.3 
|- ( ph -> W e. Word S )
Assertion wrdpmcl 
|- ( ph -> ( W o. E ) e. Word S )

Proof

Step Hyp Ref Expression
1 wrdpmcl.1 
 |-  J = ( 0 ..^ ( # ` W ) )
2 wrdpmcl.2 
 |-  ( ph -> E : J -1-1-onto-> J )
3 wrdpmcl.3 
 |-  ( ph -> W e. Word S )
4 eqidd 
 |-  ( ph -> ( # ` W ) = ( # ` W ) )
5 4 3 wrdfd 
 |-  ( ph -> W : ( 0 ..^ ( # ` W ) ) --> S )
6 f1oeq23 
 |-  ( ( J = ( 0 ..^ ( # ` W ) ) /\ J = ( 0 ..^ ( # ` W ) ) ) -> ( E : J -1-1-onto-> J <-> E : ( 0 ..^ ( # ` W ) ) -1-1-onto-> ( 0 ..^ ( # ` W ) ) ) )
7 1 1 6 mp2an 
 |-  ( E : J -1-1-onto-> J <-> E : ( 0 ..^ ( # ` W ) ) -1-1-onto-> ( 0 ..^ ( # ` W ) ) )
8 2 7 sylib 
 |-  ( ph -> E : ( 0 ..^ ( # ` W ) ) -1-1-onto-> ( 0 ..^ ( # ` W ) ) )
9 f1of 
 |-  ( E : ( 0 ..^ ( # ` W ) ) -1-1-onto-> ( 0 ..^ ( # ` W ) ) -> E : ( 0 ..^ ( # ` W ) ) --> ( 0 ..^ ( # ` W ) ) )
10 8 9 syl 
 |-  ( ph -> E : ( 0 ..^ ( # ` W ) ) --> ( 0 ..^ ( # ` W ) ) )
11 5 10 fcod 
 |-  ( ph -> ( W o. E ) : ( 0 ..^ ( # ` W ) ) --> S )
12 iswrdi 
 |-  ( ( W o. E ) : ( 0 ..^ ( # ` W ) ) --> S -> ( W o. E ) e. Word S )
13 11 12 syl 
 |-  ( ph -> ( W o. E ) e. Word S )