Theorem bj-projun 36382

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 36379	. . . . 5 ⊢ (𝐴 Proj 𝐵) = {𝑥 ∣ {𝑥} ∈ (𝐵 “ {𝐴})}
2	1	eqabri 2871	. . . 4 ⊢ (𝑥 ∈ (𝐴 Proj 𝐵) ↔ {𝑥} ∈ (𝐵 “ {𝐴}))
3		df-bj-proj 36379	. . . . 5 ⊢ (𝐴 Proj 𝐶) = {𝑥 ∣ {𝑥} ∈ (𝐶 “ {𝐴})}
4	3	eqabri 2871	. . . 4 ⊢ (𝑥 ∈ (𝐴 Proj 𝐶) ↔ {𝑥} ∈ (𝐶 “ {𝐴}))
5	2, 4	orbi12i 911	. . 3 ⊢ ((𝑥 ∈ (𝐴 Proj 𝐵) ∨ 𝑥 ∈ (𝐴 Proj 𝐶)) ↔ ({𝑥} ∈ (𝐵 “ {𝐴}) ∨ {𝑥} ∈ (𝐶 “ {𝐴})))
6		elun 4143	. . 3 ⊢ (𝑥 ∈ ((𝐴 Proj 𝐵) ∪ (𝐴 Proj 𝐶)) ↔ (𝑥 ∈ (𝐴 Proj 𝐵) ∨ 𝑥 ∈ (𝐴 Proj 𝐶)))
7		df-bj-proj 36379	. . . . 5 ⊢ (𝐴 Proj (𝐵 ∪ 𝐶)) = {𝑥 ∣ {𝑥} ∈ ((𝐵 ∪ 𝐶) “ {𝐴})}
8	7	eqabri 2871	. . . 4 ⊢ (𝑥 ∈ (𝐴 Proj (𝐵 ∪ 𝐶)) ↔ {𝑥} ∈ ((𝐵 ∪ 𝐶) “ {𝐴}))
9		imaundir 6144	. . . . 5 ⊢ ((𝐵 ∪ 𝐶) “ {𝐴}) = ((𝐵 “ {𝐴}) ∪ (𝐶 “ {𝐴}))
10	9	eleq2i 2819	. . . 4 ⊢ ({𝑥} ∈ ((𝐵 ∪ 𝐶) “ {𝐴}) ↔ {𝑥} ∈ ((𝐵 “ {𝐴}) ∪ (𝐶 “ {𝐴})))
11		elun 4143	. . . 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 2723	1 ⊢ (𝐴 Proj (𝐵 ∪ 𝐶)) = ((𝐴 Proj 𝐵) ∪ (𝐴 Proj 𝐶))

Syntax hints:   ∨ wo 844   = wceq 1533   ∈ wcel 2098   ∪ cun 3941  {csn 4623   “ cima 5672   Proj bj-cproj 36378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-bj-proj 36379

This theorem is referenced by:  bj-pr1un  36391  bj-pr2un  36405