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Theorem bj-pr1un 34212
Description: The first projection preserves unions. (Contributed by BJ, 6-Apr-2019.)
Assertion
Ref Expression
bj-pr1un pr1 (𝐴𝐵) = (pr1 𝐴 ∪ pr1 𝐵)

Proof of Theorem bj-pr1un
StepHypRef Expression
1 bj-projun 34203 . 2 (∅ Proj (𝐴𝐵)) = ((∅ Proj 𝐴) ∪ (∅ Proj 𝐵))
2 df-bj-pr1 34210 . 2 pr1 (𝐴𝐵) = (∅ Proj (𝐴𝐵))
3 df-bj-pr1 34210 . . 3 pr1 𝐴 = (∅ Proj 𝐴)
4 df-bj-pr1 34210 . . 3 pr1 𝐵 = (∅ Proj 𝐵)
53, 4uneq12i 4134 . 2 (pr1 𝐴 ∪ pr1 𝐵) = ((∅ Proj 𝐴) ∪ (∅ Proj 𝐵))
61, 2, 53eqtr4i 2851 1 pr1 (𝐴𝐵) = (pr1 𝐴 ∪ pr1 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  cun 3931  c0 4288   Proj bj-cproj 34199  pr1 bj-cpr1 34209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-bj-proj 34200  df-bj-pr1 34210
This theorem is referenced by:  bj-pr21val  34222
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