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Metamath Proof Explorer

Theorem rnghmresfn

Description: The class of non-unital ring homomorphisms restricted to subsets of non-unital rings is a function. (Contributed by AV, 4-Mar-2020)

Ref Expression
Hypotheses rnghmresfn.b 
|- ( ph -> B = ( U i^i Rng ) )
rnghmresfn.h 
|- ( ph -> H = ( RngHom |` ( B X. B ) ) )
Assertion rnghmresfn 
|- ( ph -> H Fn ( B X. B ) )

Proof

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