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Theorem bj-projun 33209
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 33206 . . . . 5 (𝐴 Proj 𝐵) = {𝑥 ∣ {𝑥} ∈ (𝐵 “ {𝐴})}
21abeq2i 2837 . . . 4 (𝑥 ∈ (𝐴 Proj 𝐵) ↔ {𝑥} ∈ (𝐵 “ {𝐴}))
3 df-bj-proj 33206 . . . . 5 (𝐴 Proj 𝐶) = {𝑥 ∣ {𝑥} ∈ (𝐶 “ {𝐴})}
43abeq2i 2837 . . . 4 (𝑥 ∈ (𝐴 Proj 𝐶) ↔ {𝑥} ∈ (𝐶 “ {𝐴}))
52, 4orbi12i 544 . . 3 ((𝑥 ∈ (𝐴 Proj 𝐵) ∨ 𝑥 ∈ (𝐴 Proj 𝐶)) ↔ ({𝑥} ∈ (𝐵 “ {𝐴}) ∨ {𝑥} ∈ (𝐶 “ {𝐴})))
6 elun 3861 . . 3 (𝑥 ∈ ((𝐴 Proj 𝐵) ∪ (𝐴 Proj 𝐶)) ↔ (𝑥 ∈ (𝐴 Proj 𝐵) ∨ 𝑥 ∈ (𝐴 Proj 𝐶)))
7 df-bj-proj 33206 . . . . 5 (𝐴 Proj (𝐵𝐶)) = {𝑥 ∣ {𝑥} ∈ ((𝐵𝐶) “ {𝐴})}
87abeq2i 2837 . . . 4 (𝑥 ∈ (𝐴 Proj (𝐵𝐶)) ↔ {𝑥} ∈ ((𝐵𝐶) “ {𝐴}))
9 imaundir 5656 . . . . 5 ((𝐵𝐶) “ {𝐴}) = ((𝐵 “ {𝐴}) ∪ (𝐶 “ {𝐴}))
109eleq2i 2795 . . . 4 ({𝑥} ∈ ((𝐵𝐶) “ {𝐴}) ↔ {𝑥} ∈ ((𝐵 “ {𝐴}) ∪ (𝐶 “ {𝐴})))
11 elun 3861 . . . 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 2721 1 (𝐴 Proj (𝐵𝐶)) = ((𝐴 Proj 𝐵) ∪ (𝐴 Proj 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wo 382   = wceq 1596  wcel 2103  cun 3678  {csn 4285  cima 5221   Proj bj-cproj 33205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-br 4761  df-opab 4821  df-cnv 5226  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-bj-proj 33206
This theorem is referenced by:  bj-pr1un  33218  bj-pr2un  33232
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