Theorem bj-pr21val 36997

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 36995	. . 3 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
2		bj-pr1eq 36986	. . 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 36987	. 2 ⊢ pr1 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) = (pr1 ⦅𝐴⦆ ∪ pr1 ({1o} × tag 𝐵))
5		bj-pr11val 36989	. . . 4 ⊢ pr1 ⦅𝐴⦆ = 𝐴
6		bj-pr1val 36988	. . . . 5 ⊢ pr1 ({1o} × tag 𝐵) = if(1o = ∅, 𝐵, ∅)
7		1n0 8406	. . . . . . 7 ⊢ 1o ≠ ∅
8	7	neii 2927	. . . . . 6 ⊢ ¬ 1o = ∅
9	8	iffalsei 4486	. . . . 5 ⊢ if(1o = ∅, 𝐵, ∅) = ∅
10	6, 9	eqtri 2752	. . . 4 ⊢ pr1 ({1o} × tag 𝐵) = ∅
11	5, 10	uneq12i 4117	. . 3 ⊢ (pr1 ⦅𝐴⦆ ∪ pr1 ({1o} × tag 𝐵)) = (𝐴 ∪ ∅)
12		un0 4345	. . 3 ⊢ (𝐴 ∪ ∅) = 𝐴
13	11, 12	eqtri 2752	. 2 ⊢ (pr1 ⦅𝐴⦆ ∪ pr1 ({1o} × tag 𝐵)) = 𝐴
14	3, 4, 13	3eqtri 2756	1 ⊢ pr1 ⦅𝐴, 𝐵⦆ = 𝐴

Syntax hints:   = wceq 1540   ∪ cun 3901  ∅c0 4284  ifcif 4476  {csn 4577   × cxp 5617  1_oc1o 8381  tag bj-ctag 36958  ⦅bj-c1upl 36981  pr₁ bj-cpr1 36984  ⦅bj-c2uple 36994

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-xp 5625  df-rel 5626  df-cnv 5627  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-suc 6313  df-1o 8388  df-bj-sngl 36950  df-bj-tag 36959  df-bj-proj 36975  df-bj-1upl 36982  df-bj-pr1 36985  df-bj-2upl 36995

This theorem is referenced by:  bj-2uplth  37005  bj-2uplex  37006