Theorem uniiunlem 3353

Description: A subset relationship useful for converting union to indexed union using dfiun2 4001 or dfiun2g 3999 and intersection to indexed intersection using dfiin2 4002. (Contributed by NM, 5-Oct-2006.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)

Assertion

Ref	Expression
uniiunlem	|- (A.x e. A B e. D -> (A.x e. A B e. C <-> {y | E.x e. A y = B} C_ C))

Distinct variable groups:   x,y   y,A   y,B   x,C

Allowed substitution hints:   A(x)   B(x)   C(y)   D(x,y)

Proof of Theorem uniiunlem

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2359	. . . . . 6 |- (y = z -> (y = B <-> z = B))
2	1	rexbidv 2635	. . . . 5 |- (y = z -> (E.x e. A y = B <-> E.x e. A z = B))
3	2	cbvabv 2472	. . . 4 |- {y | E.x e. A y = B} = {z | E.x e. A z = B}
4	3	sseq1i 3295	. . 3 |- ({y | E.x e. A y = B} C_ C <-> {z | E.x e. A z = B} C_ C)
5		r19.23v 2730	. . . . 5 |- (A.x e. A (z = B -> z e. C) <-> (E.x e. A z = B -> z e. C))
6	5	albii 1566	. . . 4 |- (A.zA.x e. A (z = B -> z e. C) <-> A.z(E.x e. A z = B -> z e. C))
7		ralcom4 2877	. . . 4 |- (A.x e. A A.z(z = B -> z e. C) <-> A.zA.x e. A (z = B -> z e. C))
8		abss 3335	. . . 4 |- ({z | E.x e. A z = B} C_ C <-> A.z(E.x e. A z = B -> z e. C))
9	6, 7, 8	3bitr4i 268	. . 3 |- (A.x e. A A.z(z = B -> z e. C) <-> {z | E.x e. A z = B} C_ C)
10	4, 9	bitr4i 243	. 2 |- ({y | E.x e. A y = B} C_ C <-> A.x e. A A.z(z = B -> z e. C))
11		nfv 1619	. . . . 5 |- F/z B e. C
12		eleq1 2413	. . . . 5 |- (z = B -> (z e. C <-> B e. C))
13	11, 12	ceqsalg 2883	. . . 4 |- (B e. D -> (A.z(z = B -> z e. C) <-> B e. C))
14	13	ralimi 2689	. . 3 |- (A.x e. A B e. D -> A.x e. A (A.z(z = B -> z e. C) <-> B e. C))
15		ralbi 2750	. . 3 |- (A.x e. A (A.z(z = B -> z e. C) <-> B e. C) -> (A.x e. A A.z(z = B -> z e. C) <-> A.x e. A B e. C))
16	14, 15	syl 15	. 2 |- (A.x e. A B e. D -> (A.x e. A A.z(z = B -> z e. C) <-> A.x e. A B e. C))
17	10, 16	syl5rbb 249	1 |- (A.x e. A B e. D -> (A.x e. A B e. C <-> {y | E.x e. A y = B} C_ C))

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540   = wceq 1642   e. wcel 1710  {cab 2339  A.wral 2614  E.wrex 2615   C_ wss 3257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259

This theorem is referenced by: (None)