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Metamath Proof Explorer

Theorem hashfundm

Description: The size of a set function is equal to the size of its domain. (Contributed by BTernaryTau, 30-Sep-2023)

Ref Expression
Assertion hashfundm 
|- ( ( F e. V /\ Fun F ) -> ( # ` F ) = ( # ` dom F ) )

Proof

Step Hyp Ref Expression
1 hashfun 
 |-  ( F e. Fin -> ( Fun F <-> ( # ` F ) = ( # ` dom F ) ) )
2 1 biimpd 
 |-  ( F e. Fin -> ( Fun F -> ( # ` F ) = ( # ` dom F ) ) )
3 2 adantld 
 |-  ( F e. Fin -> ( ( F e. V /\ Fun F ) -> ( # ` F ) = ( # ` dom F ) ) )
4 hashinf 
 |-  ( ( F e. V /\ -. F e. Fin ) -> ( # ` F ) = +oo )
5 4 3adant2 
 |-  ( ( F e. V /\ Fun F /\ -. F e. Fin ) -> ( # ` F ) = +oo )
6 fundmfibi 
 |-  ( Fun F -> ( F e. Fin <-> dom F e. Fin ) )
7 6 notbid 
 |-  ( Fun F -> ( -. F e. Fin <-> -. dom F e. Fin ) )
8 7 adantl 
 |-  ( ( F e. V /\ Fun F ) -> ( -. F e. Fin <-> -. dom F e. Fin ) )
9 dmexg 
 |-  ( F e. V -> dom F e. _V )
10 hashinf 
 |-  ( ( dom F e. _V /\ -. dom F e. Fin ) -> ( # ` dom F ) = +oo )
11 9 10 sylan 
 |-  ( ( F e. V /\ -. dom F e. Fin ) -> ( # ` dom F ) = +oo )
12 11 ex 
 |-  ( F e. V -> ( -. dom F e. Fin -> ( # ` dom F ) = +oo ) )
13 12 adantr 
 |-  ( ( F e. V /\ Fun F ) -> ( -. dom F e. Fin -> ( # ` dom F ) = +oo ) )
14 8 13 sylbid 
 |-  ( ( F e. V /\ Fun F ) -> ( -. F e. Fin -> ( # ` dom F ) = +oo ) )
15 14 3impia 
 |-  ( ( F e. V /\ Fun F /\ -. F e. Fin ) -> ( # ` dom F ) = +oo )
16 5 15 eqtr4d 
 |-  ( ( F e. V /\ Fun F /\ -. F e. Fin ) -> ( # ` F ) = ( # ` dom F ) )
17 16 3comr 
 |-  ( ( -. F e. Fin /\ F e. V /\ Fun F ) -> ( # ` F ) = ( # ` dom F ) )
18 17 3expib 
 |-  ( -. F e. Fin -> ( ( F e. V /\ Fun F ) -> ( # ` F ) = ( # ` dom F ) ) )
19 3 18 pm2.61i 
 |-  ( ( F e. V /\ Fun F ) -> ( # ` F ) = ( # ` dom F ) )