Theorem iuneq2 3904

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 3903	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶)
2		ss2iun 3903	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
3	1, 2	anim12i 338	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵) → (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶 ∧ ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵))
4		eqss 3172	. . . 4 ⊢ (𝐵 = 𝐶 ↔ (𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵))
5	4	ralbii 2483	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 ↔ ∀𝑥 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵))
6		r19.26 2603	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) ↔ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵))
7	5, 6	bitri 184	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 ↔ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵))
8		eqss 3172	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶 ↔ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶 ∧ ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵))
9	3, 7, 8	3imtr4i 201	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353  ∀wral 2455   ⊆ wss 3131  ∪ ciun 3888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-in 3137  df-ss 3144  df-iun 3890

This theorem is referenced by:  iuneq2i  3906  iuneq2dv  3909  dfmptg  5697