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Theorem bj-pr2un 33955
 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 33932 . 2 (1o Proj (𝐴𝐵)) = ((1o Proj 𝐴) ∪ (1o Proj 𝐵))
2 df-bj-pr2 33953 . 2 pr2 (𝐴𝐵) = (1o Proj (𝐴𝐵))
3 df-bj-pr2 33953 . . 3 pr2 𝐴 = (1o Proj 𝐴)
4 df-bj-pr2 33953 . . 3 pr2 𝐵 = (1o Proj 𝐵)
53, 4uneq12i 4064 . 2 (pr2 𝐴 ∪ pr2 𝐵) = ((1o Proj 𝐴) ∪ (1o Proj 𝐵))
61, 2, 53eqtr4i 2831 1 pr2 (𝐴𝐵) = (pr2 𝐴 ∪ pr2 𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1525   ∪ cun 3863  1oc1o 7953   Proj bj-cproj 33928  pr2 bj-cpr2 33952 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-rab 3116  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-br 4969  df-opab 5031  df-cnv 5458  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-bj-proj 33929  df-bj-pr2 33953 This theorem is referenced by:  bj-pr22val  33957
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