Theorem iuneq1 3717

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 3715	. . 3 ⊢ (𝐴 ⊆ 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑥 ∈ 𝐵 𝐶)
2		iunss1 3715	. . 3 ⊢ (𝐵 ⊆ 𝐴 → ∪ 𝑥 ∈ 𝐵 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶)
3	1, 2	anim12i 331	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴) → (∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑥 ∈ 𝐵 𝐶 ∧ ∪ 𝑥 ∈ 𝐵 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶))
4		eqss 3025	. 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
5		eqss 3025	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶 ↔ (∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑥 ∈ 𝐵 𝐶 ∧ ∪ 𝑥 ∈ 𝐵 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶))
6	3, 4, 5	3imtr4i 199	1 ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶)

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1285   ⊆ wss 2984  ∪ ciun 3704

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-in 2990  df-ss 2997  df-iun 3706

This theorem is referenced by:  iuneq1d  3727  iununir  3785  iunsuc  4210  rdgisuc1  6079  rdg0  6082  oasuc  6155  omsuc  6163