Theorem iunin1f 29374

Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4573 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.) (Revised by Thierry Arnoux, 2-May-2020.)

Hypothesis

Ref	Expression
iunin1f.1	|- F/_ x C

Assertion

Ref	Expression
iunin1f	|- U_ x e. A ( B i^i C ) = ( U_ x e. A B i^i C )

Proof of Theorem iunin1f

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iunin1f.1	. . . . . 6 |- F/_ x C
2	1	nfcri 2758	. . . . 5 |- F/ x y e. C
3	2	r19.41 3090	. . . 4 |- ( E. x e. A ( y e. B /\ y e. C ) <-> ( E. x e. A y e. B /\ y e. C ) )
4		elin 3796	. . . . 5 |- ( y e. ( B i^i C ) <-> ( y e. B /\ y e. C ) )
5	4	rexbii 3041	. . . 4 |- ( E. x e. A y e. ( B i^i C ) <-> E. x e. A ( y e. B /\ y e. C ) )
6		eliun 4524	. . . . 5 |- ( y e. U_ x e. A B <-> E. x e. A y e. B )
7	6	anbi1i 731	. . . 4 |- ( ( y e. U_ x e. A B /\ y e. C ) <-> ( E. x e. A y e. B /\ y e. C ) )
8	3, 5, 7	3bitr4i 292	. . 3 |- ( E. x e. A y e. ( B i^i C ) <-> ( y e. U_ x e. A B /\ y e. C ) )
9		eliun 4524	. . 3 |- ( y e. U_ x e. A ( B i^i C ) <-> E. x e. A y e. ( B i^i C ) )
10		elin 3796	. . 3 |- ( y e. ( U_ x e. A B i^i C ) <-> ( y e. U_ x e. A B /\ y e. C ) )
11	8, 9, 10	3bitr4i 292	. 2 |- ( y e. U_ x e. A ( B i^i C ) <-> y e. ( U_ x e. A B i^i C ) )
12	11	eqriv 2619	1 |- U_ x e. A ( B i^i C ) = ( U_ x e. A B i^i C )

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   F/_wnfc 2751   E.wrex 2913   i^i cin 3573   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-in 3581  df-iun 4522

This theorem is referenced by:  esum2dlem  30154