Theorem bj-projun 34301

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 34298	. . . . 5 ⊢ (𝐴 Proj 𝐵) = {𝑥 ∣ {𝑥} ∈ (𝐵 “ {𝐴})}
2	1	abeq2i 2948	. . . 4 ⊢ (𝑥 ∈ (𝐴 Proj 𝐵) ↔ {𝑥} ∈ (𝐵 “ {𝐴}))
3		df-bj-proj 34298	. . . . 5 ⊢ (𝐴 Proj 𝐶) = {𝑥 ∣ {𝑥} ∈ (𝐶 “ {𝐴})}
4	3	abeq2i 2948	. . . 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 4124	. . 3 ⊢ (𝑥 ∈ ((𝐴 Proj 𝐵) ∪ (𝐴 Proj 𝐶)) ↔ (𝑥 ∈ (𝐴 Proj 𝐵) ∨ 𝑥 ∈ (𝐴 Proj 𝐶)))
7		df-bj-proj 34298	. . . . 5 ⊢ (𝐴 Proj (𝐵 ∪ 𝐶)) = {𝑥 ∣ {𝑥} ∈ ((𝐵 ∪ 𝐶) “ {𝐴})}
8	7	abeq2i 2948	. . . 4 ⊢ (𝑥 ∈ (𝐴 Proj (𝐵 ∪ 𝐶)) ↔ {𝑥} ∈ ((𝐵 ∪ 𝐶) “ {𝐴}))
9		imaundir 6003	. . . . 5 ⊢ ((𝐵 ∪ 𝐶) “ {𝐴}) = ((𝐵 “ {𝐴}) ∪ (𝐶 “ {𝐴}))
10	9	eleq2i 2904	. . . 4 ⊢ ({𝑥} ∈ ((𝐵 ∪ 𝐶) “ {𝐴}) ↔ {𝑥} ∈ ((𝐵 “ {𝐴}) ∪ (𝐶 “ {𝐴})))
11		elun 4124	. . . 4 ⊢ ({𝑥} ∈ ((𝐵 “ {𝐴}) ∪ (𝐶 “ {𝐴})) ↔ ({𝑥} ∈ (𝐵 “ {𝐴}) ∨ {𝑥} ∈ (𝐶 “ {𝐴})))
12	8, 10, 11	3bitri 299	. . 3 ⊢ (𝑥 ∈ (𝐴 Proj (𝐵 ∪ 𝐶)) ↔ ({𝑥} ∈ (𝐵 “ {𝐴}) ∨ {𝑥} ∈ (𝐶 “ {𝐴})))
13	5, 6, 12	3bitr4ri 306	. 2 ⊢ (𝑥 ∈ (𝐴 Proj (𝐵 ∪ 𝐶)) ↔ 𝑥 ∈ ((𝐴 Proj 𝐵) ∪ (𝐴 Proj 𝐶)))
14	13	eqriv 2818	1 ⊢ (𝐴 Proj (𝐵 ∪ 𝐶)) = ((𝐴 Proj 𝐵) ∪ (𝐴 Proj 𝐶))

Syntax hints:   ∨ wo 843   = wceq 1533   ∈ wcel 2110   ∪ cun 3933  {csn 4560   “ cima 5552   Proj bj-cproj 34297

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-br 5059  df-opab 5121  df-cnv 5557  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-bj-proj 34298

This theorem is referenced by:  bj-pr1un  34310  bj-pr2un  34324