Theorem uniiunlem 3022

Description: A subset relationship useful for converting union to indexed union using dfiun2 or dfiun2g and intersection to indexed intersection using dfiin2 . (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 2043	. . . . . 6 |- (y = z -> (y = B <-> z = B))
2	1	rexbidv 2321	. . . . 5 |- (y = z -> (E.x e. A y = B <-> E.x e. A z = B))
3	2	cbvabv 2158	. . . 4 |- {y | E.x e. A y = B } = {z | E.x e. A z = B }
4	3	sseq1i 2963	. . 3 |- ( {y | E.x e. A y = B } C_ C <-> {z | E.x e. A z = B } C_ C)
5		r19.23v 2419	. . . . 5 |- (A.x e. A (z = B -> z e. C) <-> (E.x e. A z = B -> z e. C))
6	5	albii 1356	. . . 4 |- (A.zA.x e. A (z = B -> z e. C) <-> A.z(E.x e. A z = B -> z e. C))
7		ralcom4 2570	. . . 4 |- (A.x e. A A.z(z = B -> z e. C) <-> A.zA.x e. A (z = B -> z e. C))
8		abss 3003	. . . 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 201	. . 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 176	. 2 |- ( {y | E.x e. A y = B } C_ C <-> A.x e. A A.z(z = B -> z e. C))
11		nfv 1418	. . . . 5 |- F/z B e. C
12		eleq1 2097	. . . . 5 |- (z = B -> (z e. C <-> B e. C))
13	11, 12	ceqsalg 2576	. . . 4 |- (B e. D -> (A.z(z = B -> z e. C) <-> B e. C))
14	13	ralimi 2378	. . 3 |- (A.x e. A B e. D -> A.x e. A (A.z(z = B -> z e. C) <-> B e. C))
15		ralbi 2439	. . 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 14	. 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 182	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 98  A.wal 1240   = wceq 1242   e. wcel 1390   {cab 2023  A.wral 2300  E.wrex 2301   C_ wss 2911

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925

This theorem is referenced by: (None)