Theorem iuneq2 3647

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

Assertion

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

Proof of Theorem iuneq2

Step	Hyp	Ref	Expression
1		ss2iun 3646	. . 3 ⊢ (∀x ∈ A B ⊆ 𝐶 → ∪ x ∈ A B ⊆ ∪ x ∈ A 𝐶)
2		ss2iun 3646	. . 3 ⊢ (∀x ∈ A 𝐶 ⊆ B → ∪ x ∈ A 𝐶 ⊆ ∪ x ∈ A B)
3	1, 2	anim12i 321	. 2 ⊢ ((∀x ∈ A B ⊆ 𝐶 ∧ ∀x ∈ A 𝐶 ⊆ B) → (∪ x ∈ A B ⊆ ∪ x ∈ A 𝐶 ∧ ∪ x ∈ A 𝐶 ⊆ ∪ x ∈ A B))
4		eqss 2937	. . . 4 ⊢ (B = 𝐶 ↔ (B ⊆ 𝐶 ∧ 𝐶 ⊆ B))
5	4	ralbii 2308	. . 3 ⊢ (∀x ∈ A B = 𝐶 ↔ ∀x ∈ A (B ⊆ 𝐶 ∧ 𝐶 ⊆ B))
6		r19.26 2419	. . 3 ⊢ (∀x ∈ A (B ⊆ 𝐶 ∧ 𝐶 ⊆ B) ↔ (∀x ∈ A B ⊆ 𝐶 ∧ ∀x ∈ A 𝐶 ⊆ B))
7	5, 6	bitri 173	. 2 ⊢ (∀x ∈ A B = 𝐶 ↔ (∀x ∈ A B ⊆ 𝐶 ∧ ∀x ∈ A 𝐶 ⊆ B))
8		eqss 2937	. 2 ⊢ (∪ x ∈ A B = ∪ x ∈ A 𝐶 ↔ (∪ x ∈ A B ⊆ ∪ x ∈ A 𝐶 ∧ ∪ x ∈ A 𝐶 ⊆ ∪ x ∈ A B))
9	3, 7, 8	3imtr4i 190	1 ⊢ (∀x ∈ A B = 𝐶 → ∪ x ∈ A B = ∪ x ∈ A 𝐶)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1228  ∀wral 2284   ⊆ wss 2894  ∪ ciun 3631

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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-in 2901  df-ss 2908  df-iun 3633

This theorem is referenced by:  iuneq2i  3649  iuneq2dv  3652  dfmptg  5267