Theorem bj-projun 37027

Description: The class projection on a given component preserves unions. (Contributed by BJ, 6-Apr-2019.)

Assertion

Ref	Expression
bj-projun	⊢ (𝐴 Proj (𝐵 ∪ 𝐶)) = ((𝐴 Proj 𝐵) ∪ (𝐴 Proj 𝐶))

Proof of Theorem bj-projun

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-bj-proj 37024	. . . . 5 ⊢ (𝐴 Proj 𝐵) = {𝑥 ∣ {𝑥} ∈ (𝐵 “ {𝐴})}
2	1	eqabri 2874	. . . 4 ⊢ (𝑥 ∈ (𝐴 Proj 𝐵) ↔ {𝑥} ∈ (𝐵 “ {𝐴}))
3		df-bj-proj 37024	. . . . 5 ⊢ (𝐴 Proj 𝐶) = {𝑥 ∣ {𝑥} ∈ (𝐶 “ {𝐴})}
4	3	eqabri 2874	. . . 4 ⊢ (𝑥 ∈ (𝐴 Proj 𝐶) ↔ {𝑥} ∈ (𝐶 “ {𝐴}))
5	2, 4	orbi12i 914	. . 3 ⊢ ((𝑥 ∈ (𝐴 Proj 𝐵) ∨ 𝑥 ∈ (𝐴 Proj 𝐶)) ↔ ({𝑥} ∈ (𝐵 “ {𝐴}) ∨ {𝑥} ∈ (𝐶 “ {𝐴})))
6		elun 4103	. . 3 ⊢ (𝑥 ∈ ((𝐴 Proj 𝐵) ∪ (𝐴 Proj 𝐶)) ↔ (𝑥 ∈ (𝐴 Proj 𝐵) ∨ 𝑥 ∈ (𝐴 Proj 𝐶)))
7		df-bj-proj 37024	. . . . 5 ⊢ (𝐴 Proj (𝐵 ∪ 𝐶)) = {𝑥 ∣ {𝑥} ∈ ((𝐵 ∪ 𝐶) “ {𝐴})}
8	7	eqabri 2874	. . . 4 ⊢ (𝑥 ∈ (𝐴 Proj (𝐵 ∪ 𝐶)) ↔ {𝑥} ∈ ((𝐵 ∪ 𝐶) “ {𝐴}))
9		imaundir 6097	. . . . 5 ⊢ ((𝐵 ∪ 𝐶) “ {𝐴}) = ((𝐵 “ {𝐴}) ∪ (𝐶 “ {𝐴}))
10	9	eleq2i 2823	. . . 4 ⊢ ({𝑥} ∈ ((𝐵 ∪ 𝐶) “ {𝐴}) ↔ {𝑥} ∈ ((𝐵 “ {𝐴}) ∪ (𝐶 “ {𝐴})))
11		elun 4103	. . . 4 ⊢ ({𝑥} ∈ ((𝐵 “ {𝐴}) ∪ (𝐶 “ {𝐴})) ↔ ({𝑥} ∈ (𝐵 “ {𝐴}) ∨ {𝑥} ∈ (𝐶 “ {𝐴})))
12	8, 10, 11	3bitri 297	. . 3 ⊢ (𝑥 ∈ (𝐴 Proj (𝐵 ∪ 𝐶)) ↔ ({𝑥} ∈ (𝐵 “ {𝐴}) ∨ {𝑥} ∈ (𝐶 “ {𝐴})))
13	5, 6, 12	3bitr4ri 304	. 2 ⊢ (𝑥 ∈ (𝐴 Proj (𝐵 ∪ 𝐶)) ↔ 𝑥 ∈ ((𝐴 Proj 𝐵) ∪ (𝐴 Proj 𝐶)))
14	13	eqriv 2728	1 ⊢ (𝐴 Proj (𝐵 ∪ 𝐶)) = ((𝐴 Proj 𝐵) ∪ (𝐴 Proj 𝐶))

Syntax hints:   ∨ wo 847   = wceq 1541   ∈ wcel 2111   ∪ cun 3900  {csn 4576   “ cima 5619   Proj bj-cproj 37023

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-cnv 5624  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-bj-proj 37024

This theorem is referenced by:  bj-pr1un  37036  bj-pr2un  37050