Theorem bj-pr21val 36952

Description: Value of the first projection of a couple. (Contributed by BJ, 6-Oct-2018.)

Assertion

Ref	Expression
bj-pr21val	⊢ pr1 ⦅𝐴, 𝐵⦆ = 𝐴

Proof of Theorem bj-pr21val

Step	Hyp	Ref	Expression
1		df-bj-2upl 36950	. . 3 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
2		bj-pr1eq 36941	. . 3 ⊢ (⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) → pr1 ⦅𝐴, 𝐵⦆ = pr1 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)))
3	1, 2	ax-mp 5	. 2 ⊢ pr1 ⦅𝐴, 𝐵⦆ = pr1 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
4		bj-pr1un 36942	. 2 ⊢ pr1 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) = (pr1 ⦅𝐴⦆ ∪ pr1 ({1o} × tag 𝐵))
5		bj-pr11val 36944	. . . 4 ⊢ pr1 ⦅𝐴⦆ = 𝐴
6		bj-pr1val 36943	. . . . 5 ⊢ pr1 ({1o} × tag 𝐵) = if(1o = ∅, 𝐵, ∅)
7		1n0 8494	. . . . . . 7 ⊢ 1o ≠ ∅
8	7	neii 2933	. . . . . 6 ⊢ ¬ 1o = ∅
9	8	iffalsei 4508	. . . . 5 ⊢ if(1o = ∅, 𝐵, ∅) = ∅
10	6, 9	eqtri 2757	. . . 4 ⊢ pr1 ({1o} × tag 𝐵) = ∅
11	5, 10	uneq12i 4139	. . 3 ⊢ (pr1 ⦅𝐴⦆ ∪ pr1 ({1o} × tag 𝐵)) = (𝐴 ∪ ∅)
12		un0 4367	. . 3 ⊢ (𝐴 ∪ ∅) = 𝐴
13	11, 12	eqtri 2757	. 2 ⊢ (pr1 ⦅𝐴⦆ ∪ pr1 ({1o} × tag 𝐵)) = 𝐴
14	3, 4, 13	3eqtri 2761	1 ⊢ pr1 ⦅𝐴, 𝐵⦆ = 𝐴

Syntax hints:   = wceq 1539   ∪ cun 3922  ∅c0 4306  ifcif 4498  {csn 4599   × cxp 5649  1_oc1o 8467  tag bj-ctag 36913  ⦅bj-c1upl 36936  pr₁ bj-cpr1 36939  ⦅bj-c2uple 36949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5263  ax-nul 5273  ax-pr 5399

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3414  df-v 3459  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-sn 4600  df-pr 4602  df-op 4606  df-br 5117  df-opab 5179  df-xp 5657  df-rel 5658  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-suc 6355  df-1o 8474  df-bj-sngl 36905  df-bj-tag 36914  df-bj-proj 36930  df-bj-1upl 36937  df-bj-pr1 36940  df-bj-2upl 36950

This theorem is referenced by:  bj-2uplth  36960  bj-2uplex  36961