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Theorem bj-pr21val 32976
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 32974 . . 3 𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐵))
2 bj-pr1eq 32965 . . 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 32966 . 2 pr1 (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐵)) = (pr1𝐴⦆ ∪ pr1 ({1𝑜} × tag 𝐵))
5 bj-pr11val 32968 . . . 4 pr1𝐴⦆ = 𝐴
6 bj-pr1val 32967 . . . . 5 pr1 ({1𝑜} × tag 𝐵) = if(1𝑜 = ∅, 𝐵, ∅)
7 1n0 7560 . . . . . . 7 1𝑜 ≠ ∅
87neii 2793 . . . . . 6 ¬ 1𝑜 = ∅
98iffalsei 4087 . . . . 5 if(1𝑜 = ∅, 𝐵, ∅) = ∅
106, 9eqtri 2642 . . . 4 pr1 ({1𝑜} × tag 𝐵) = ∅
115, 10uneq12i 3757 . . 3 (pr1𝐴⦆ ∪ pr1 ({1𝑜} × tag 𝐵)) = (𝐴 ∪ ∅)
12 un0 3958 . . 3 (𝐴 ∪ ∅) = 𝐴
1311, 12eqtri 2642 . 2 (pr1𝐴⦆ ∪ pr1 ({1𝑜} × tag 𝐵)) = 𝐴
143, 4, 133eqtri 2646 1 pr1𝐴, 𝐵⦆ = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1481  cun 3565  c0 3907  ifcif 4077  {csn 4168   × cxp 5102  1𝑜c1o 7538  tag bj-ctag 32937  bj-c1upl 32960  pr1 bj-cpr1 32963  bj-c2uple 32973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-opab 4704  df-xp 5110  df-rel 5111  df-cnv 5112  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-suc 5717  df-1o 7545  df-bj-sngl 32929  df-bj-tag 32938  df-bj-proj 32954  df-bj-1upl 32961  df-bj-pr1 32964  df-bj-2upl 32974
This theorem is referenced by:  bj-2uplth  32984  bj-2uplex  32985
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