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Theorem bj-pr2un 36362
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 36339 . 2 (1o Proj (𝐴𝐵)) = ((1o Proj 𝐴) ∪ (1o Proj 𝐵))
2 df-bj-pr2 36360 . 2 pr2 (𝐴𝐵) = (1o Proj (𝐴𝐵))
3 df-bj-pr2 36360 . . 3 pr2 𝐴 = (1o Proj 𝐴)
4 df-bj-pr2 36360 . . 3 pr2 𝐵 = (1o Proj 𝐵)
53, 4uneq12i 4161 . 2 (pr2 𝐴 ∪ pr2 𝐵) = ((1o Proj 𝐴) ∪ (1o Proj 𝐵))
61, 2, 53eqtr4i 2769 1 pr2 (𝐴𝐵) = (pr2 𝐴 ∪ pr2 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  cun 3946  1oc1o 8465   Proj bj-cproj 36335  pr2 bj-cpr2 36359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-bj-proj 36336  df-bj-pr2 36360
This theorem is referenced by:  bj-pr22val  36364
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