Theorem iuneq2 2578

Description: Equality theorem for indexed union.

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 2577	. . 3 |- (A.x e. A B (_ C -> U_x e. A B (_ U_x e. A C)
2		ss2iun 2577	. . 3 |- (A.x e. A C (_ B -> U_x e. A C (_ U_x e. A B)
3	1, 2	anim12i 333	. 2 |- ((A.x e. A B (_ C /\ A.x e. A C (_ B) -> (U_x e. A B (_ U_x e. A C /\ U_x e. A C (_ U_x e. A B))
4		eqss 2077	. . . 4 |- (B = C <-> (B (_ C /\ C (_ B))
5	4	ralbii 1667	. . 3 |- (A.x e. A B = C <-> A.x e. A (B (_ C /\ C (_ B))
6		r19.26 1750	. . 3 |- (A.x e. A (B (_ C /\ C (_ B) <-> (A.x e. A B (_ C /\ A.x e. A C (_ B))
7	5, 6	bitr 173	. 2 |- (A.x e. A B = C <-> (A.x e. A B (_ C /\ A.x e. A C (_ B))
8		eqss 2077	. 2 |- (U_x e. A B = U_x e. A C <-> (U_x e. A B (_ U_x e. A C /\ U_x e. A C (_ U_x e. A B))
9	3, 7, 8	3imtr4 219	1 |- (A.x e. A B = C -> U_x e. A B = U_x e. A C)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956  A.wral 1645   (_ wss 2047  U_ciun 2566

This theorem is referenced by:  iuneq2i 2580  iuneq2dv 2582  abianfplem 3961  oa0r 4173  om0r 4174  om1r 4177  oe1m 4179  oaass 4195  oarec 4196  omass 4211  oaabs 4252  r1val3 4679  kmlem11 4775  cardiun 4859

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-rex 1650  df-v 1812  df-in 2051  df-ss 2053  df-iun 2568

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