Theorem bj-pr22val 33007

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 32999	. . . 4 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐵))
2		bj-pr2eq 33004	. . . 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 33005	. . 3 ⊢ pr2 (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐵)) = (pr2 ⦅𝐴⦆ ∪ pr2 ({1𝑜} × tag 𝐵))
5	3, 4	eqtri 2644	. 2 ⊢ pr2 ⦅𝐴, 𝐵⦆ = (pr2 ⦅𝐴⦆ ∪ pr2 ({1𝑜} × tag 𝐵))
6		df-bj-1upl 32986	. . . . 5 ⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴)
7		bj-pr2eq 33004	. . . . 5 ⊢ (⦅𝐴⦆ = ({∅} × tag 𝐴) → pr2 ⦅𝐴⦆ = pr2 ({∅} × tag 𝐴))
8	6, 7	ax-mp 5	. . . 4 ⊢ pr2 ⦅𝐴⦆ = pr2 ({∅} × tag 𝐴)
9		bj-pr2val 33006	. . . 4 ⊢ pr2 ({∅} × tag 𝐴) = if(∅ = 1𝑜, 𝐴, ∅)
10		1n0 7575	. . . . . 6 ⊢ 1𝑜 ≠ ∅
11	10	nesymi 2851	. . . . 5 ⊢ ¬ ∅ = 1𝑜
12	11	iffalsei 4096	. . . 4 ⊢ if(∅ = 1𝑜, 𝐴, ∅) = ∅
13	8, 9, 12	3eqtri 2648	. . 3 ⊢ pr2 ⦅𝐴⦆ = ∅
14		bj-pr2val 33006	. . . 4 ⊢ pr2 ({1𝑜} × tag 𝐵) = if(1𝑜 = 1𝑜, 𝐵, ∅)
15		eqid 2622	. . . . 5 ⊢ 1𝑜 = 1𝑜
16	15	iftruei 4093	. . . 4 ⊢ if(1𝑜 = 1𝑜, 𝐵, ∅) = 𝐵
17	14, 16	eqtri 2644	. . 3 ⊢ pr2 ({1𝑜} × tag 𝐵) = 𝐵
18	13, 17	uneq12i 3765	. 2 ⊢ (pr2 ⦅𝐴⦆ ∪ pr2 ({1𝑜} × tag 𝐵)) = (∅ ∪ 𝐵)
19		uncom 3757	. . 3 ⊢ (∅ ∪ 𝐵) = (𝐵 ∪ ∅)
20		un0 3967	. . 3 ⊢ (𝐵 ∪ ∅) = 𝐵
21	19, 20	eqtri 2644	. 2 ⊢ (∅ ∪ 𝐵) = 𝐵
22	5, 18, 21	3eqtri 2648	1 ⊢ pr2 ⦅𝐴, 𝐵⦆ = 𝐵

Syntax hints:   = wceq 1483   ∪ cun 3572  ∅c0 3915  ifcif 4086  {csn 4177   × cxp 5112  1_𝑜c1o 7553  tag bj-ctag 32962  ⦅bj-c1upl 32985  ⦅bj-c2uple 32998  pr₂ bj-cpr2 33002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-suc 5729  df-1o 7560  df-bj-sngl 32954  df-bj-tag 32963  df-bj-proj 32979  df-bj-1upl 32986  df-bj-2upl 32999  df-bj-pr2 33003

This theorem is referenced by:  bj-2uplth  33009  bj-2uplex  33010