Theorem iuneq1 3983

Description: Equality theorem for indexed union. (Contributed by NM, 27-Jun-1998.)

Assertion

Ref	Expression
iuneq1	⊢ (A = B → ∪x ∈ A C = ∪x ∈ B C)

Distinct variable groups:   x,A   x,B

Allowed substitution hint:   C(x)

Proof of Theorem iuneq1

Step	Hyp	Ref	Expression
1		iunss1 3981	. . 3 ⊢ (A ⊆ B → ∪x ∈ A C ⊆ ∪x ∈ B C)
2		iunss1 3981	. . 3 ⊢ (B ⊆ A → ∪x ∈ B C ⊆ ∪x ∈ A C)
3	1, 2	anim12i 549	. 2 ⊢ ((A ⊆ B ∧ B ⊆ A) → (∪x ∈ A C ⊆ ∪x ∈ B C ∧ ∪x ∈ B C ⊆ ∪x ∈ A C))
4		eqss 3288	. 2 ⊢ (A = B ↔ (A ⊆ B ∧ B ⊆ A))
5		eqss 3288	. 2 ⊢ (∪x ∈ A C = ∪x ∈ B C ↔ (∪x ∈ A C ⊆ ∪x ∈ B C ∧ ∪x ∈ B C ⊆ ∪x ∈ A C))
6	3, 4, 5	3imtr4i 257	1 ⊢ (A = B → ∪x ∈ A C = ∪x ∈ B C)

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ⊆ wss 3258  ∪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:  iuneq1d  3993  iununi  4051