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Theorem bj-pr21val 34450
Description: Value of the first projection of a couple. (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-pr21val pr1𝐴, 𝐵⦆ = 𝐴

Proof of Theorem bj-pr21val
StepHypRef Expression
1 df-bj-2upl 34448 . . 3 𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
2 bj-pr1eq 34439 . . 3 (⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) → pr1𝐴, 𝐵⦆ = pr1 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)))
31, 2ax-mp 5 . 2 pr1𝐴, 𝐵⦆ = pr1 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
4 bj-pr1un 34440 . 2 pr1 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) = (pr1𝐴⦆ ∪ pr1 ({1o} × tag 𝐵))
5 bj-pr11val 34442 . . . 4 pr1𝐴⦆ = 𝐴
6 bj-pr1val 34441 . . . . 5 pr1 ({1o} × tag 𝐵) = if(1o = ∅, 𝐵, ∅)
7 1n0 8106 . . . . . . 7 1o ≠ ∅
87neii 2992 . . . . . 6 ¬ 1o = ∅
98iffalsei 4438 . . . . 5 if(1o = ∅, 𝐵, ∅) = ∅
106, 9eqtri 2824 . . . 4 pr1 ({1o} × tag 𝐵) = ∅
115, 10uneq12i 4091 . . 3 (pr1𝐴⦆ ∪ pr1 ({1o} × tag 𝐵)) = (𝐴 ∪ ∅)
12 un0 4301 . . 3 (𝐴 ∪ ∅) = 𝐴
1311, 12eqtri 2824 . 2 (pr1𝐴⦆ ∪ pr1 ({1o} × tag 𝐵)) = 𝐴
143, 4, 133eqtri 2828 1 pr1𝐴, 𝐵⦆ = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  cun 3882  c0 4246  ifcif 4428  {csn 4528   × cxp 5521  1oc1o 8082  tag bj-ctag 34411  bj-c1upl 34434  pr1 bj-cpr1 34437  bj-c2uple 34447
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-xp 5529  df-rel 5530  df-cnv 5531  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-suc 6169  df-1o 8089  df-bj-sngl 34403  df-bj-tag 34412  df-bj-proj 34428  df-bj-1upl 34435  df-bj-pr1 34438  df-bj-2upl 34448
This theorem is referenced by:  bj-2uplth  34458  bj-2uplex  34459
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