Theorem bj-projun 37746

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 37743	. . . . 5 ⊢ (𝐴 Proj 𝐵) = {𝑥 ∣ {𝑥} ∈ (𝐵 “ {𝐴})}
2	1	eqabri 2904	. . . 4 ⊢ (𝑥 ∈ (𝐴 Proj 𝐵) ↔ {𝑥} ∈ (𝐵 “ {𝐴}))
3		df-bj-proj 37743	. . . . 5 ⊢ (𝐴 Proj 𝐶) = {𝑥 ∣ {𝑥} ∈ (𝐶 “ {𝐴})}
4	3	eqabri 2904	. . . 4 ⊢ (𝑥 ∈ (𝐴 Proj 𝐶) ↔ {𝑥} ∈ (𝐶 “ {𝐴}))
5	2, 4	orbi12i 928	. . 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 37743	. . . . 5 ⊢ (𝐴 Proj (𝐵 ∪ 𝐶)) = {𝑥 ∣ {𝑥} ∈ ((𝐵 ∪ 𝐶) “ {𝐴})}
8	7	eqabri 2904	. . . 4 ⊢ (𝑥 ∈ (𝐴 Proj (𝐵 ∪ 𝐶)) ↔ {𝑥} ∈ ((𝐵 ∪ 𝐶) “ {𝐴}))
9		imaundir 6146	. . . . 5 ⊢ ((𝐵 ∪ 𝐶) “ {𝐴}) = ((𝐵 “ {𝐴}) ∪ (𝐶 “ {𝐴}))
10	9	eleq2i 2854	. . . 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 300	. . 3 ⊢ (𝑥 ∈ (𝐴 Proj (𝐵 ∪ 𝐶)) ↔ ({𝑥} ∈ (𝐵 “ {𝐴}) ∨ {𝑥} ∈ (𝐶 “ {𝐴})))
13	5, 6, 12	3bitr4ri 307	. 2 ⊢ (𝑥 ∈ (𝐴 Proj (𝐵 ∪ 𝐶)) ↔ 𝑥 ∈ ((𝐴 Proj 𝐵) ∪ (𝐴 Proj 𝐶)))
14	13	eqriv 2759	1 ⊢ (𝐴 Proj (𝐵 ∪ 𝐶)) = ((𝐴 Proj 𝐵) ∪ (𝐴 Proj 𝐶))

This proof depends on syntax axioms:   ∨ wo 861   = wceq 1570   ∈ wcel 2145   ∪ cun 3900  {csn 4587   “ cima 5662   Proj bj-cproj 37742

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-12 2215  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-bj-proj 37743

This theorem is used by:  bj-pr1un  37755  bj-pr2un  37769