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Theorem bj-projun 36530
Description: The class projection on a given component preserves unions. (Contributed by BJ, 6-Apr-2019.)
Assertion
Ref Expression
bj-projun (𝐴 Proj (𝐵𝐶)) = ((𝐴 Proj 𝐵) ∪ (𝐴 Proj 𝐶))

Proof of Theorem bj-projun
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-bj-proj 36527 . . . . 5 (𝐴 Proj 𝐵) = {𝑥 ∣ {𝑥} ∈ (𝐵 “ {𝐴})}
21eqabri 2869 . . . 4 (𝑥 ∈ (𝐴 Proj 𝐵) ↔ {𝑥} ∈ (𝐵 “ {𝐴}))
3 df-bj-proj 36527 . . . . 5 (𝐴 Proj 𝐶) = {𝑥 ∣ {𝑥} ∈ (𝐶 “ {𝐴})}
43eqabri 2869 . . . 4 (𝑥 ∈ (𝐴 Proj 𝐶) ↔ {𝑥} ∈ (𝐶 “ {𝐴}))
52, 4orbi12i 912 . . 3 ((𝑥 ∈ (𝐴 Proj 𝐵) ∨ 𝑥 ∈ (𝐴 Proj 𝐶)) ↔ ({𝑥} ∈ (𝐵 “ {𝐴}) ∨ {𝑥} ∈ (𝐶 “ {𝐴})))
6 elun 4141 . . 3 (𝑥 ∈ ((𝐴 Proj 𝐵) ∪ (𝐴 Proj 𝐶)) ↔ (𝑥 ∈ (𝐴 Proj 𝐵) ∨ 𝑥 ∈ (𝐴 Proj 𝐶)))
7 df-bj-proj 36527 . . . . 5 (𝐴 Proj (𝐵𝐶)) = {𝑥 ∣ {𝑥} ∈ ((𝐵𝐶) “ {𝐴})}
87eqabri 2869 . . . 4 (𝑥 ∈ (𝐴 Proj (𝐵𝐶)) ↔ {𝑥} ∈ ((𝐵𝐶) “ {𝐴}))
9 imaundir 6150 . . . . 5 ((𝐵𝐶) “ {𝐴}) = ((𝐵 “ {𝐴}) ∪ (𝐶 “ {𝐴}))
109eleq2i 2817 . . . 4 ({𝑥} ∈ ((𝐵𝐶) “ {𝐴}) ↔ {𝑥} ∈ ((𝐵 “ {𝐴}) ∪ (𝐶 “ {𝐴})))
11 elun 4141 . . . 4 ({𝑥} ∈ ((𝐵 “ {𝐴}) ∪ (𝐶 “ {𝐴})) ↔ ({𝑥} ∈ (𝐵 “ {𝐴}) ∨ {𝑥} ∈ (𝐶 “ {𝐴})))
128, 10, 113bitri 296 . . 3 (𝑥 ∈ (𝐴 Proj (𝐵𝐶)) ↔ ({𝑥} ∈ (𝐵 “ {𝐴}) ∨ {𝑥} ∈ (𝐶 “ {𝐴})))
135, 6, 123bitr4ri 303 . 2 (𝑥 ∈ (𝐴 Proj (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴 Proj 𝐵) ∪ (𝐴 Proj 𝐶)))
1413eqriv 2722 1 (𝐴 Proj (𝐵𝐶)) = ((𝐴 Proj 𝐵) ∪ (𝐴 Proj 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wo 845   = wceq 1533  wcel 2098  cun 3937  {csn 4624  cima 5675   Proj bj-cproj 36526
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 2166  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-cnv 5680  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-bj-proj 36527
This theorem is referenced by:  bj-pr1un  36539  bj-pr2un  36553
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