Theorem bj-pr22val 36555

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

Assertion

Ref	Expression
bj-pr22val	⊢ pr2 ⦅𝐴, 𝐵⦆ = 𝐵

Proof of Theorem bj-pr22val

Step	Hyp	Ref	Expression
1		df-bj-2upl 36547	. . . 4 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
2		bj-pr2eq 36552	. . . 4 ⊢ (⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) → pr2 ⦅𝐴, 𝐵⦆ = pr2 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)))
3	1, 2	ax-mp 5	. . 3 ⊢ pr2 ⦅𝐴, 𝐵⦆ = pr2 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
4		bj-pr2un 36553	. . 3 ⊢ pr2 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) = (pr2 ⦅𝐴⦆ ∪ pr2 ({1o} × tag 𝐵))
5	3, 4	eqtri 2753	. 2 ⊢ pr2 ⦅𝐴, 𝐵⦆ = (pr2 ⦅𝐴⦆ ∪ pr2 ({1o} × tag 𝐵))
6		df-bj-1upl 36534	. . . . 5 ⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴)
7		bj-pr2eq 36552	. . . . 5 ⊢ (⦅𝐴⦆ = ({∅} × tag 𝐴) → pr2 ⦅𝐴⦆ = pr2 ({∅} × tag 𝐴))
8	6, 7	ax-mp 5	. . . 4 ⊢ pr2 ⦅𝐴⦆ = pr2 ({∅} × tag 𝐴)
9		bj-pr2val 36554	. . . 4 ⊢ pr2 ({∅} × tag 𝐴) = if(∅ = 1o, 𝐴, ∅)
10		1n0 8507	. . . . . 6 ⊢ 1o ≠ ∅
11	10	nesymi 2988	. . . . 5 ⊢ ¬ ∅ = 1o
12	11	iffalsei 4534	. . . 4 ⊢ if(∅ = 1o, 𝐴, ∅) = ∅
13	8, 9, 12	3eqtri 2757	. . 3 ⊢ pr2 ⦅𝐴⦆ = ∅
14		bj-pr2val 36554	. . . 4 ⊢ pr2 ({1o} × tag 𝐵) = if(1o = 1o, 𝐵, ∅)
15		eqid 2725	. . . . 5 ⊢ 1o = 1o
16	15	iftruei 4531	. . . 4 ⊢ if(1o = 1o, 𝐵, ∅) = 𝐵
17	14, 16	eqtri 2753	. . 3 ⊢ pr2 ({1o} × tag 𝐵) = 𝐵
18	13, 17	uneq12i 4154	. 2 ⊢ (pr2 ⦅𝐴⦆ ∪ pr2 ({1o} × tag 𝐵)) = (∅ ∪ 𝐵)
19		0un 4388	. 2 ⊢ (∅ ∪ 𝐵) = 𝐵
20	5, 18, 19	3eqtri 2757	1 ⊢ pr2 ⦅𝐴, 𝐵⦆ = 𝐵

Syntax hints:   = wceq 1533   ∪ cun 3937  ∅c0 4318  ifcif 4524  {csn 4624   × cxp 5670  1_oc1o 8478  tag bj-ctag 36510  ⦅bj-c1upl 36533  ⦅bj-c2uple 36546  pr₂ bj-cpr2 36550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-xp 5678  df-rel 5679  df-cnv 5680  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-suc 6370  df-1o 8485  df-bj-sngl 36502  df-bj-tag 36511  df-bj-proj 36527  df-bj-1upl 36534  df-bj-2upl 36547  df-bj-pr2 36551

This theorem is referenced by:  bj-2uplth  36557  bj-2uplex  36558