Theorem eliuni 4526

Description: Membership in an indexed union, one way. (Contributed by JJ, 27-Jul-2021.)

Hypothesis

Ref	Expression
eliuni.1	|- ( x = A -> B = C )

Assertion

Ref	Expression
eliuni	|- ( ( A e. D /\ E e. C ) -> E e. U_ x e. D B )

Distinct variable groups:   x, A   x, C   x, D   x, E

Allowed substitution hint:   B( x)

Proof of Theorem eliuni

Step	Hyp	Ref	Expression
1		eliuni.1	. . . 4 |- ( x = A -> B = C )
2	1	eleq2d 2687	. . 3 |- ( x = A -> ( E e. B <-> E e. C ) )
3	2	rspcev 3309	. 2 |- ( ( A e. D /\ E e. C ) -> E. x e. D E e. B )
4		eliun 4524	. 2 |- ( E e. U_ x e. D B <-> E. x e. D E e. B )
5	3, 4	sylibr 224	1 |- ( ( A e. D /\ E e. C ) -> E e. U_ x e. D B )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   U_ciun 4520

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-rex 2918  df-v 3202  df-iun 4522

This theorem is referenced by:  oeordi  7667  fseqdom  8849  cfsmolem  9092  axdc3lem2  9273  prmreclem5  15624  efgs1b  18149  lbsextlem2  19159  pmatcoe1fsupp  20506  vitalilem2  23378  cnrefiisplem  40055