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Theorem bj-pr11val 34460
 Description: Value of the first projection of a monuple. (Contributed by BJ, 6-Apr-2019.)
Assertion
Ref Expression
bj-pr11val pr1𝐴⦆ = 𝐴

Proof of Theorem bj-pr11val
StepHypRef Expression
1 df-bj-1upl 34453 . . 3 𝐴⦆ = ({∅} × tag 𝐴)
2 bj-pr1eq 34457 . . 3 (⦅𝐴⦆ = ({∅} × tag 𝐴) → pr1𝐴⦆ = pr1 ({∅} × tag 𝐴))
31, 2ax-mp 5 . 2 pr1𝐴⦆ = pr1 ({∅} × tag 𝐴)
4 bj-pr1val 34459 . 2 pr1 ({∅} × tag 𝐴) = if(∅ = ∅, 𝐴, ∅)
5 eqid 2798 . . 3 ∅ = ∅
65iftruei 4432 . 2 if(∅ = ∅, 𝐴, ∅) = 𝐴
73, 4, 63eqtri 2825 1 pr1𝐴⦆ = 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538  ∅c0 4243  ifcif 4425  {csn 4525   × cxp 5518  tag bj-ctag 34429  ⦅bj-c1upl 34452  pr1 bj-cpr1 34455 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5032  df-opab 5094  df-xp 5526  df-rel 5527  df-cnv 5528  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-bj-sngl 34421  df-bj-tag 34430  df-bj-proj 34446  df-bj-1upl 34453  df-bj-pr1 34456 This theorem is referenced by:  bj-1uplth  34462  bj-1uplex  34463  bj-pr21val  34468
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