Theorem iuneq2 2006

Description: Equality theorem for indexed union.

Assertion

Ref	Expression
iuneq2	⊢ (∀x ∈ A B = C → ∪x ∈ A B = ∪x ∈ A C)

Proof of Theorem iuneq2

Step	Hyp	Ref	Expression
1		ss2iun 2005	. . 3 ⊢ (∀x ∈ A B ⊆ C → ∪x ∈ A B ⊆ ∪x ∈ A C)
2		ss2iun 2005	. . 3 ⊢ (∀x ∈ A C ⊆ B → ∪x ∈ A C ⊆ ∪x ∈ A B)
3	1, 2	anim12i 268	. 2 ⊢ ((∀x ∈ A B ⊆ C ∧ ∀x ∈ A C ⊆ B) → (∪x ∈ A B ⊆ ∪x ∈ A C ∧ ∪x ∈ A C ⊆ ∪x ∈ A B))
4		eqss 1516	. . . 4 ⊢ (B = C ↔ (B ⊆ C ∧ C ⊆ B))
5	4	biral 1223	. . 3 ⊢ (∀x ∈ A B = C ↔ ∀x ∈ A (B ⊆ C ∧ C ⊆ B))
6		r19.26 1289	. . 3 ⊢ (∀x ∈ A (B ⊆ C ∧ C ⊆ B) ↔ (∀x ∈ A B ⊆ C ∧ ∀x ∈ A C ⊆ B))
7	5, 6	bitr 151	. 2 ⊢ (∀x ∈ A B = C ↔ (∀x ∈ A B ⊆ C ∧ ∀x ∈ A C ⊆ B))
8		eqss 1516	. 2 ⊢ (∪x ∈ A B = ∪x ∈ A C ↔ (∪x ∈ A B ⊆ ∪x ∈ A C ∧ ∪x ∈ A C ⊆ ∪x ∈ A B))
9	3, 7, 8	3imtr4 192	1 ⊢ (∀x ∈ A B = C → ∪x ∈ A B = ∪x ∈ A C)

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091  ∀wral 1201   ⊆ wss 1487  ∪ciun 1994

This theorem is referenced by:  iuneq2i 2008  iuneq2dv 2010  abianfplem 2999  oa0r 3141  om0r 3142  om1r 3145  oe1m 3147  oaass 3163  r1val3 3523  kmlem10 3589  cardiun 3665

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-in 1491  df-ss 1492  df-iun 1996

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