Theorem srhmsubclem1 42073

Description: Lemma 1 for srhmsubc 42076. (Contributed by AV, 19-Feb-2020.)

Hypotheses

Ref	Expression
srhmsubc.s	|- A. r e. S r e. Ring
srhmsubc.c	|- C = ( U i^i S )

Assertion

Ref	Expression
srhmsubclem1	|- ( X e. C -> X e. ( U i^i Ring ) )

Distinct variable groups:   S, r   X, r

Allowed substitution hints:   C( r)   U( r)

Proof of Theorem srhmsubclem1

Step	Hyp	Ref	Expression
1		eleq1 2689	. . . 4 |- ( r = X -> ( r e. Ring <-> X e. Ring ) )
2		srhmsubc.s	. . . 4 |- A. r e. S r e. Ring
3	1, 2	vtoclri 3283	. . 3 |- ( X e. S -> X e. Ring )
4	3	anim2i 593	. 2 |- ( ( X e. U /\ X e. S ) -> ( X e. U /\ X e. Ring ) )
5		srhmsubc.c	. . 3 |- C = ( U i^i S )
6	5	elin2 3801	. 2 |- ( X e. C <-> ( X e. U /\ X e. S ) )
7		elin 3796	. 2 |- ( X e. ( U i^i Ring ) <-> ( X e. U /\ X e. Ring ) )
8	4, 6, 7	3imtr4i 281	1 |- ( X e. C -> X e. ( U i^i Ring ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   i^i cin 3573   Ringcrg 18547

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-v 3202  df-in 3581

This theorem is referenced by:  srhmsubclem2  42074  srhmsubcALTVlem1  42092