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Theorem bj-projun 35397
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 35394 . . . . 5 (𝐴 Proj 𝐵) = {𝑥 ∣ {𝑥} ∈ (𝐵 “ {𝐴})}
21abeq2i 2875 . . . 4 (𝑥 ∈ (𝐴 Proj 𝐵) ↔ {𝑥} ∈ (𝐵 “ {𝐴}))
3 df-bj-proj 35394 . . . . 5 (𝐴 Proj 𝐶) = {𝑥 ∣ {𝑥} ∈ (𝐶 “ {𝐴})}
43abeq2i 2875 . . . 4 (𝑥 ∈ (𝐴 Proj 𝐶) ↔ {𝑥} ∈ (𝐶 “ {𝐴}))
52, 4orbi12i 914 . . 3 ((𝑥 ∈ (𝐴 Proj 𝐵) ∨ 𝑥 ∈ (𝐴 Proj 𝐶)) ↔ ({𝑥} ∈ (𝐵 “ {𝐴}) ∨ {𝑥} ∈ (𝐶 “ {𝐴})))
6 elun 4107 . . 3 (𝑥 ∈ ((𝐴 Proj 𝐵) ∪ (𝐴 Proj 𝐶)) ↔ (𝑥 ∈ (𝐴 Proj 𝐵) ∨ 𝑥 ∈ (𝐴 Proj 𝐶)))
7 df-bj-proj 35394 . . . . 5 (𝐴 Proj (𝐵𝐶)) = {𝑥 ∣ {𝑥} ∈ ((𝐵𝐶) “ {𝐴})}
87abeq2i 2875 . . . 4 (𝑥 ∈ (𝐴 Proj (𝐵𝐶)) ↔ {𝑥} ∈ ((𝐵𝐶) “ {𝐴}))
9 imaundir 6102 . . . . 5 ((𝐵𝐶) “ {𝐴}) = ((𝐵 “ {𝐴}) ∪ (𝐶 “ {𝐴}))
109eleq2i 2830 . . . 4 ({𝑥} ∈ ((𝐵𝐶) “ {𝐴}) ↔ {𝑥} ∈ ((𝐵 “ {𝐴}) ∪ (𝐶 “ {𝐴})))
11 elun 4107 . . . 4 ({𝑥} ∈ ((𝐵 “ {𝐴}) ∪ (𝐶 “ {𝐴})) ↔ ({𝑥} ∈ (𝐵 “ {𝐴}) ∨ {𝑥} ∈ (𝐶 “ {𝐴})))
128, 10, 113bitri 297 . . 3 (𝑥 ∈ (𝐴 Proj (𝐵𝐶)) ↔ ({𝑥} ∈ (𝐵 “ {𝐴}) ∨ {𝑥} ∈ (𝐶 “ {𝐴})))
135, 6, 123bitr4ri 304 . 2 (𝑥 ∈ (𝐴 Proj (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴 Proj 𝐵) ∪ (𝐴 Proj 𝐶)))
1413eqriv 2735 1 (𝐴 Proj (𝐵𝐶)) = ((𝐴 Proj 𝐵) ∪ (𝐴 Proj 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wo 846   = wceq 1542  wcel 2107  cun 3907  {csn 4585  cima 5635   Proj bj-cproj 35393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-bj-proj 35394
This theorem is referenced by:  bj-pr1un  35406  bj-pr2un  35420
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