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Metamath Proof Explorer

Theorem rhmresfn

Description: The class of unital ring homomorphisms restricted to subsets of unital rings is a function. (Contributed by AV, 10-Mar-2020)

Ref Expression
Hypotheses rhmresfn.b 
|- ( ph -> B = ( U i^i Ring ) )
rhmresfn.h 
|- ( ph -> H = ( RingHom |` ( B X. B ) ) )
Assertion rhmresfn 
|- ( ph -> H Fn ( B X. B ) )

Proof

Step Hyp Ref Expression
1 rhmresfn.b 
 |-  ( ph -> B = ( U i^i Ring ) )
2 rhmresfn.h 
 |-  ( ph -> H = ( RingHom |` ( B X. B ) ) )
3 rhmfn 
 |-  RingHom Fn ( Ring X. Ring )
4 inss2 
 |-  ( U i^i Ring ) C_ Ring
5 1 4 eqsstrdi 
 |-  ( ph -> B C_ Ring )
6 xpss12 
 |-  ( ( B C_ Ring /\ B C_ Ring ) -> ( B X. B ) C_ ( Ring X. Ring ) )
7 5 5 6 syl2anc 
 |-  ( ph -> ( B X. B ) C_ ( Ring X. Ring ) )
8 fnssres 
 |-  ( ( RingHom Fn ( Ring X. Ring ) /\ ( B X. B ) C_ ( Ring X. Ring ) ) -> ( RingHom |` ( B X. B ) ) Fn ( B X. B ) )
9 3 7 8 sylancr 
 |-  ( ph -> ( RingHom |` ( B X. B ) ) Fn ( B X. B ) )
10 2 fneq1d 
 |-  ( ph -> ( H Fn ( B X. B ) <-> ( RingHom |` ( B X. B ) ) Fn ( B X. B ) ) )
11 9 10 mpbird 
 |-  ( ph -> H Fn ( B X. B ) )