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Theorem bj-pr1un 36613
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 36604 . 2 (∅ Proj (𝐴𝐵)) = ((∅ Proj 𝐴) ∪ (∅ Proj 𝐵))
2 df-bj-pr1 36611 . 2 pr1 (𝐴𝐵) = (∅ Proj (𝐴𝐵))
3 df-bj-pr1 36611 . . 3 pr1 𝐴 = (∅ Proj 𝐴)
4 df-bj-pr1 36611 . . 3 pr1 𝐵 = (∅ Proj 𝐵)
53, 4uneq12i 4158 . 2 (pr1 𝐴 ∪ pr1 𝐵) = ((∅ Proj 𝐴) ∪ (∅ Proj 𝐵))
61, 2, 53eqtr4i 2763 1 pr1 (𝐴𝐵) = (pr1 𝐴 ∪ pr1 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  cun 3942  c0 4322   Proj bj-cproj 36600  pr1 bj-cpr1 36610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-bj-proj 36601  df-bj-pr1 36611
This theorem is referenced by:  bj-pr21val  36623
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