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Theorem bj-pr2un 32198
 Description: The second projection preserves unions. (Contributed by BJ, 6-Apr-2019.)
Assertion
Ref Expression
bj-pr2un pr2 (𝐴𝐵) = (pr2 𝐴 ∪ pr2 𝐵)

Proof of Theorem bj-pr2un
StepHypRef Expression
1 bj-projun 32175 . 2 (1𝑜 Proj (𝐴𝐵)) = ((1𝑜 Proj 𝐴) ∪ (1𝑜 Proj 𝐵))
2 df-bj-pr2 32196 . 2 pr2 (𝐴𝐵) = (1𝑜 Proj (𝐴𝐵))
3 df-bj-pr2 32196 . . 3 pr2 𝐴 = (1𝑜 Proj 𝐴)
4 df-bj-pr2 32196 . . 3 pr2 𝐵 = (1𝑜 Proj 𝐵)
53, 4uneq12i 3727 . 2 (pr2 𝐴 ∪ pr2 𝐵) = ((1𝑜 Proj 𝐴) ∪ (1𝑜 Proj 𝐵))
61, 2, 53eqtr4i 2642 1 pr2 (𝐴𝐵) = (pr2 𝐴 ∪ pr2 𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∪ cun 3538  1𝑜c1o 7440   Proj bj-cproj 32171  pr2 bj-cpr2 32195 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-bj-proj 32172  df-bj-pr2 32196 This theorem is referenced by:  bj-pr22val  32200
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