Metamath Proof Explorer

Theorem hashreshashfun

Description: The number of elements of a finite function expressed by a restriction. (Contributed by AV, 15-Dec-2021)

Ref Expression
Assertion hashreshashfun 
|- ( ( Fun A /\ A e. Fin /\ B C_ dom A ) -> ( # ` A ) = ( ( # ` ( A |` B ) ) + ( # ` ( dom A \ B ) ) ) )

Proof

Step Hyp Ref Expression
1 simp1 
 |-  ( ( Fun A /\ A e. Fin /\ B C_ dom A ) -> Fun A )
2 hashfun 
 |-  ( A e. Fin -> ( Fun A <-> ( # ` A ) = ( # ` dom A ) ) )
3 2 3ad2ant2 
 |-  ( ( Fun A /\ A e. Fin /\ B C_ dom A ) -> ( Fun A <-> ( # ` A ) = ( # ` dom A ) ) )
4 1 3 mpbid 
 |-  ( ( Fun A /\ A e. Fin /\ B C_ dom A ) -> ( # ` A ) = ( # ` dom A ) )
5 dmfi 
 |-  ( A e. Fin -> dom A e. Fin )
6 5 anim1i 
 |-  ( ( A e. Fin /\ B C_ dom A ) -> ( dom A e. Fin /\ B C_ dom A ) )
7 6 3adant1 
 |-  ( ( Fun A /\ A e. Fin /\ B C_ dom A ) -> ( dom A e. Fin /\ B C_ dom A ) )
8 hashssdif 
 |-  ( ( dom A e. Fin /\ B C_ dom A ) -> ( # ` ( dom A \ B ) ) = ( ( # ` dom A ) - ( # ` B ) ) )
9 7 8 syl 
 |-  ( ( Fun A /\ A e. Fin /\ B C_ dom A ) -> ( # ` ( dom A \ B ) ) = ( ( # ` dom A ) - ( # ` B ) ) )
10 9 oveq2d 
 |-  ( ( Fun A /\ A e. Fin /\ B C_ dom A ) -> ( ( # ` B ) + ( # ` ( dom A \ B ) ) ) = ( ( # ` B ) + ( ( # ` dom A ) - ( # ` B ) ) ) )
11 ssfi 
 |-  ( ( dom A e. Fin /\ B C_ dom A ) -> B e. Fin )
12 11 ex 
 |-  ( dom A e. Fin -> ( B C_ dom A -> B e. Fin ) )
13 hashcl 
 |-  ( B e. Fin -> ( # ` B ) e. NN0 )
14 13 nn0cnd 
 |-  ( B e. Fin -> ( # ` B ) e. CC )
15 12 14 syl6 
 |-  ( dom A e. Fin -> ( B C_ dom A -> ( # ` B ) e. CC ) )
16 5 15 syl 
 |-  ( A e. Fin -> ( B C_ dom A -> ( # ` B ) e. CC ) )
17 16 imp 
 |-  ( ( A e. Fin /\ B C_ dom A ) -> ( # ` B ) e. CC )
18 hashcl 
 |-  ( dom A e. Fin -> ( # ` dom A ) e. NN0 )
19 5 18 syl 
 |-  ( A e. Fin -> ( # ` dom A ) e. NN0 )
20 19 nn0cnd 
 |-  ( A e. Fin -> ( # ` dom A ) e. CC )
21 20 adantr 
 |-  ( ( A e. Fin /\ B C_ dom A ) -> ( # ` dom A ) e. CC )
22 17 21 jca 
 |-  ( ( A e. Fin /\ B C_ dom A ) -> ( ( # ` B ) e. CC /\ ( # ` dom A ) e. CC ) )
23 22 3adant1 
 |-  ( ( Fun A /\ A e. Fin /\ B C_ dom A ) -> ( ( # ` B ) e. CC /\ ( # ` dom A ) e. CC ) )
24 pncan3 
 |-  ( ( ( # ` B ) e. CC /\ ( # ` dom A ) e. CC ) -> ( ( # ` B ) + ( ( # ` dom A ) - ( # ` B ) ) ) = ( # ` dom A ) )
25 23 24 syl 
 |-  ( ( Fun A /\ A e. Fin /\ B C_ dom A ) -> ( ( # ` B ) + ( ( # ` dom A ) - ( # ` B ) ) ) = ( # ` dom A ) )
26 10 25 eqtr2d 
 |-  ( ( Fun A /\ A e. Fin /\ B C_ dom A ) -> ( # ` dom A ) = ( ( # ` B ) + ( # ` ( dom A \ B ) ) ) )
27 hashres 
 |-  ( ( Fun A /\ A e. Fin /\ B C_ dom A ) -> ( # ` ( A |` B ) ) = ( # ` B ) )
28 27 eqcomd 
 |-  ( ( Fun A /\ A e. Fin /\ B C_ dom A ) -> ( # ` B ) = ( # ` ( A |` B ) ) )
29 28 oveq1d 
 |-  ( ( Fun A /\ A e. Fin /\ B C_ dom A ) -> ( ( # ` B ) + ( # ` ( dom A \ B ) ) ) = ( ( # ` ( A |` B ) ) + ( # ` ( dom A \ B ) ) ) )
30 4 26 29 3eqtrd 
 |-  ( ( Fun A /\ A e. Fin /\ B C_ dom A ) -> ( # ` A ) = ( ( # ` ( A |` B ) ) + ( # ` ( dom A \ B ) ) ) )