Theorem bj-pr21val 37537

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 37535	. . 3 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
2		bj-pr1eq 37526	. . 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 37527	. 2 ⊢ pr1 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) = (pr1 ⦅𝐴⦆ ∪ pr1 ({1o} × tag 𝐵))
5		bj-pr11val 37529	. . . 4 ⊢ pr1 ⦅𝐴⦆ = 𝐴
6		bj-pr1val 37528	. . . . 5 ⊢ pr1 ({1o} × tag 𝐵) = if(1o = ∅, 𝐵, ∅)
7		1n0 8472	. . . . . . 7 ⊢ 1o ≠ ∅
8	7	neii 2966	. . . . . 6 ⊢ ¬ 1o = ∅
9	8	iffalsei 4502	. . . . 5 ⊢ if(1o = ∅, 𝐵, ∅) = ∅
10	6, 9	eqtri 2792	. . . 4 ⊢ pr1 ({1o} × tag 𝐵) = ∅
11	5, 10	uneq12i 4128	. . 3 ⊢ (pr1 ⦅𝐴⦆ ∪ pr1 ({1o} × tag 𝐵)) = (𝐴 ∪ ∅)
12		un0 4358	. . 3 ⊢ (𝐴 ∪ ∅) = 𝐴
13	11, 12	eqtri 2792	. 2 ⊢ (pr1 ⦅𝐴⦆ ∪ pr1 ({1o} × tag 𝐵)) = 𝐴
14	3, 4, 13	3eqtri 2796	1 ⊢ pr1 ⦅𝐴, 𝐵⦆ = 𝐴

Syntax hints:   = wceq 1567   ∪ cun 3911  ∅c0 4294  ifcif 4492  {csn 4594   × cxp 5660  1_oc1o 8446  tag bj-ctag 37498  ⦅bj-c1upl 37521  pr₁ bj-cpr1 37524  ⦅bj-c2uple 37534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-xp 5668  df-rel 5669  df-cnv 5670  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-suc 6367  df-1o 8453  df-bj-sngl 37490  df-bj-tag 37499  df-bj-proj 37515  df-bj-1upl 37522  df-bj-pr1 37525  df-bj-2upl 37535

This theorem is referenced by:  bj-2uplth  37545  bj-2uplex  37546