Theorem ss2iun 3663

Description: Subclass theorem for indexed union. (Contributed by NM, 26-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
ss2iun	|- (A.x e. A B C_ C -> U_x e. A B C_ U_x e. A C)

Proof of Theorem ss2iun

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 2933	. . . . 5 |- (B C_ C -> (y e. B -> y e. C))
2	1	ralimi 2378	. . . 4 |- (A.x e. A B C_ C -> A.x e. A (y e. B -> y e. C))
3		rexim 2407	. . . 4 |- (A.x e. A (y e. B -> y e. C) -> (E.x e. A y e. B -> E.x e. A y e. C))
4	2, 3	syl 14	. . 3 |- (A.x e. A B C_ C -> (E.x e. A y e. B -> E.x e. A y e. C))
5		eliun 3652	. . 3 |- (y e. U_x e. A B <-> E.x e. A y e. B)
6		eliun 3652	. . 3 |- (y e. U_x e. A C <-> E.x e. A y e. C)
7	4, 5, 6	3imtr4g 194	. 2 |- (A.x e. A B C_ C -> (y e. U_x e. A B -> y e. U_x e. A C))
8	7	ssrdv 2945	1 |- (A.x e. A B C_ C -> U_x e. A B C_ U_x e. A C)

Syntax hints:   -> wi 4   e. wcel 1390  A.wral 2300  E.wrex 2301   C_ wss 2911   U_ciun 3648

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  df-iun 3650

This theorem is referenced by:  iuneq2  3664