Theorem iuneq2 3986

Description: Equality theorem for indexed union. (Contributed by NM, 22-Oct-2003.)

Assertion

Ref	Expression
iuneq2	|- (A.x e. A B = C -> U_x e. A B = U_x e. A C)

Proof of Theorem iuneq2

Step	Hyp	Ref	Expression
1		ss2iun 3985	. . 3 |- (A.x e. A B C_ C -> U_x e. A B C_ U_x e. A C)
2		ss2iun 3985	. . 3 |- (A.x e. A C C_ B -> U_x e. A C C_ U_x e. A B)
3	1, 2	anim12i 549	. 2 |- ((A.x e. A B C_ C /\ A.x e. A C C_ B) -> (U_x e. A B C_ U_x e. A C /\ U_x e. A C C_ U_x e. A B))
4		eqss 3288	. . . 4 |- (B = C <-> (B C_ C /\ C C_ B))
5	4	ralbii 2639	. . 3 |- (A.x e. A B = C <-> A.x e. A (B C_ C /\ C C_ B))
6		r19.26 2747	. . 3 |- (A.x e. A (B C_ C /\ C C_ B) <-> (A.x e. A B C_ C /\ A.x e. A C C_ B))
7	5, 6	bitri 240	. 2 |- (A.x e. A B = C <-> (A.x e. A B C_ C /\ A.x e. A C C_ B))
8		eqss 3288	. 2 |- (U_x e. A B = U_x e. A C <-> (U_x e. A B C_ U_x e. A C /\ U_x e. A C C_ U_x e. A B))
9	3, 7, 8	3imtr4i 257	1 |- (A.x e. A B = C -> U_x e. A B = U_x e. A C)

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642  A.wral 2615   C_ wss 3258  U_ciun 3970

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 2479  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-iun 3972

This theorem is referenced by:  iuneq2i  3988  iuneq2dv  3991