Theorem rhmresel 42010

Description: An element of the unital ring homomorphisms restricted to a subset of unital rings is a unital ring homomorphism. (Contributed by AV, 10-Mar-2020.)

Hypothesis

Ref	Expression
rhmresel.h	|- ( ph -> H = ( RingHom |` ( B X. B ) ) )

Assertion

Ref	Expression
rhmresel	|- ( ( ph /\ ( X e. B /\ Y e. B ) /\ F e. ( X H Y ) ) -> F e. ( X RingHom Y ) )

Proof of Theorem rhmresel

Step	Hyp	Ref	Expression
1		rhmresel.h	. . . . . 6 |- ( ph -> H = ( RingHom |` ( B X. B ) ) )
2	1	adantr 481	. . . . 5 |- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> H = ( RingHom |` ( B X. B ) ) )
3	2	oveqd 6667	. . . 4 |- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( X H Y ) = ( X ( RingHom |` ( B X. B ) ) Y ) )
4		ovres 6800	. . . . 5 |- ( ( X e. B /\ Y e. B ) -> ( X ( RingHom |` ( B X. B ) ) Y ) = ( X RingHom Y ) )
5	4	adantl 482	. . . 4 |- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( X ( RingHom |` ( B X. B ) ) Y ) = ( X RingHom Y ) )
6	3, 5	eqtrd 2656	. . 3 |- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( X H Y ) = ( X RingHom Y ) )
7	6	eleq2d 2687	. 2 |- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( F e. ( X H Y ) <-> F e. ( X RingHom Y ) ) )
8	7	biimp3a 1432	1 |- ( ( ph /\ ( X e. B /\ Y e. B ) /\ F e. ( X H Y ) ) -> F e. ( X RingHom Y ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   X. cxp 5112   |` cres 5116  (class class class)co 6650   RingHom crh 18712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-res 5126  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  rhmsubcsetclem2  42022  rhmsubcrngclem2  42028