Theorem ssiun2sf 29378

Description: Subset relationship for an indexed union. (Contributed by Thierry Arnoux, 31-Dec-2016.)

Hypotheses

Ref	Expression
ssiun2sf.1	|- F/_ x A
ssiun2sf.2	|- F/_ x C
ssiun2sf.3	|- F/_ x D
ssiun2sf.4	|- ( x = C -> B = D )

Assertion

Ref	Expression
ssiun2sf	|- ( C e. A -> D C_ U_ x e. A B )

Proof of Theorem ssiun2sf

Step	Hyp	Ref	Expression
1		ssiun2sf.2	. . 3 |- F/_ x C
2		ssiun2sf.1	. . . . 5 |- F/_ x A
3	1, 2	nfel 2777	. . . 4 |- F/ x C e. A
4		ssiun2sf.3	. . . . 5 |- F/_ x D
5		nfiu1 4550	. . . . 5 |- F/_ x U_ x e. A B
6	4, 5	nfss 3596	. . . 4 |- F/ x D C_ U_ x e. A B
7	3, 6	nfim 1825	. . 3 |- F/ x ( C e. A -> D C_ U_ x e. A B )
8		eleq1 2689	. . . 4 |- ( x = C -> ( x e. A <-> C e. A ) )
9		ssiun2sf.4	. . . . 5 |- ( x = C -> B = D )
10	9	sseq1d 3632	. . . 4 |- ( x = C -> ( B C_ U_ x e. A B <-> D C_ U_ x e. A B ) )
11	8, 10	imbi12d 334	. . 3 |- ( x = C -> ( ( x e. A -> B C_ U_ x e. A B ) <-> ( C e. A -> D C_ U_ x e. A B ) ) )
12		ssiun2 4563	. . 3 |- ( x e. A -> B C_ U_ x e. A B )
13	1, 7, 11, 12	vtoclgf 3264	. 2 |- ( C e. A -> ( C e. A -> D C_ U_ x e. A B ) )
14	13	pm2.43i 52	1 |- ( C e. A -> D C_ U_ x e. A B )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   F/_wnfc 2751   C_ wss 3574   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-ss 3588  df-iun 4522

This theorem is referenced by:  iundisj2f  29403  esum2dlem  30154  voliune  30292  volfiniune  30293