Theorem bj-pr21val 35199

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 35197	. . 3 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
2		bj-pr1eq 35188	. . 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 35189	. 2 ⊢ pr1 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) = (pr1 ⦅𝐴⦆ ∪ pr1 ({1o} × tag 𝐵))
5		bj-pr11val 35191	. . . 4 ⊢ pr1 ⦅𝐴⦆ = 𝐴
6		bj-pr1val 35190	. . . . 5 ⊢ pr1 ({1o} × tag 𝐵) = if(1o = ∅, 𝐵, ∅)
7		1n0 8309	. . . . . . 7 ⊢ 1o ≠ ∅
8	7	neii 2947	. . . . . 6 ⊢ ¬ 1o = ∅
9	8	iffalsei 4475	. . . . 5 ⊢ if(1o = ∅, 𝐵, ∅) = ∅
10	6, 9	eqtri 2768	. . . 4 ⊢ pr1 ({1o} × tag 𝐵) = ∅
11	5, 10	uneq12i 4100	. . 3 ⊢ (pr1 ⦅𝐴⦆ ∪ pr1 ({1o} × tag 𝐵)) = (𝐴 ∪ ∅)
12		un0 4330	. . 3 ⊢ (𝐴 ∪ ∅) = 𝐴
13	11, 12	eqtri 2768	. 2 ⊢ (pr1 ⦅𝐴⦆ ∪ pr1 ({1o} × tag 𝐵)) = 𝐴
14	3, 4, 13	3eqtri 2772	1 ⊢ pr1 ⦅𝐴, 𝐵⦆ = 𝐴

Syntax hints:   = wceq 1542   ∪ cun 3890  ∅c0 4262  ifcif 4465  {csn 4567   × cxp 5588  1_oc1o 8281  tag bj-ctag 35160  ⦅bj-c1upl 35183  pr₁ bj-cpr1 35186  ⦅bj-c2uple 35196

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-br 5080  df-opab 5142  df-xp 5596  df-rel 5597  df-cnv 5598  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-suc 6271  df-1o 8288  df-bj-sngl 35152  df-bj-tag 35161  df-bj-proj 35177  df-bj-1upl 35184  df-bj-pr1 35187  df-bj-2upl 35197

This theorem is referenced by:  bj-2uplth  35207  bj-2uplex  35208