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Theorem bj-projun 32650
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 32647 . . . . 5 (𝐴 Proj 𝐵) = {𝑥 ∣ {𝑥} ∈ (𝐵 “ {𝐴})}
21abeq2i 2732 . . . 4 (𝑥 ∈ (𝐴 Proj 𝐵) ↔ {𝑥} ∈ (𝐵 “ {𝐴}))
3 df-bj-proj 32647 . . . . 5 (𝐴 Proj 𝐶) = {𝑥 ∣ {𝑥} ∈ (𝐶 “ {𝐴})}
43abeq2i 2732 . . . 4 (𝑥 ∈ (𝐴 Proj 𝐶) ↔ {𝑥} ∈ (𝐶 “ {𝐴}))
52, 4orbi12i 543 . . 3 ((𝑥 ∈ (𝐴 Proj 𝐵) ∨ 𝑥 ∈ (𝐴 Proj 𝐶)) ↔ ({𝑥} ∈ (𝐵 “ {𝐴}) ∨ {𝑥} ∈ (𝐶 “ {𝐴})))
6 elun 3733 . . 3 (𝑥 ∈ ((𝐴 Proj 𝐵) ∪ (𝐴 Proj 𝐶)) ↔ (𝑥 ∈ (𝐴 Proj 𝐵) ∨ 𝑥 ∈ (𝐴 Proj 𝐶)))
7 df-bj-proj 32647 . . . . 5 (𝐴 Proj (𝐵𝐶)) = {𝑥 ∣ {𝑥} ∈ ((𝐵𝐶) “ {𝐴})}
87abeq2i 2732 . . . 4 (𝑥 ∈ (𝐴 Proj (𝐵𝐶)) ↔ {𝑥} ∈ ((𝐵𝐶) “ {𝐴}))
9 imaundir 5507 . . . . 5 ((𝐵𝐶) “ {𝐴}) = ((𝐵 “ {𝐴}) ∪ (𝐶 “ {𝐴}))
109eleq2i 2690 . . . 4 ({𝑥} ∈ ((𝐵𝐶) “ {𝐴}) ↔ {𝑥} ∈ ((𝐵 “ {𝐴}) ∪ (𝐶 “ {𝐴})))
11 elun 3733 . . . 4 ({𝑥} ∈ ((𝐵 “ {𝐴}) ∪ (𝐶 “ {𝐴})) ↔ ({𝑥} ∈ (𝐵 “ {𝐴}) ∨ {𝑥} ∈ (𝐶 “ {𝐴})))
128, 10, 113bitri 286 . . 3 (𝑥 ∈ (𝐴 Proj (𝐵𝐶)) ↔ ({𝑥} ∈ (𝐵 “ {𝐴}) ∨ {𝑥} ∈ (𝐶 “ {𝐴})))
135, 6, 123bitr4ri 293 . 2 (𝑥 ∈ (𝐴 Proj (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴 Proj 𝐵) ∪ (𝐴 Proj 𝐶)))
1413eqriv 2618 1 (𝐴 Proj (𝐵𝐶)) = ((𝐴 Proj 𝐵) ∪ (𝐴 Proj 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wo 383   = wceq 1480  wcel 1987  cun 3554  {csn 4150  cima 5079   Proj bj-cproj 32646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3188  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-br 4616  df-opab 4676  df-cnv 5084  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-bj-proj 32647
This theorem is referenced by:  bj-pr1un  32659  bj-pr2un  32673
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