Theorem bj-pr22val 32200

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 32192	. . . 4 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐵))
2		bj-pr2eq 32197	. . . 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 32198	. . 3 ⊢ pr2 (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐵)) = (pr2 ⦅𝐴⦆ ∪ pr2 ({1𝑜} × tag 𝐵))
5	3, 4	eqtri 2632	. 2 ⊢ pr2 ⦅𝐴, 𝐵⦆ = (pr2 ⦅𝐴⦆ ∪ pr2 ({1𝑜} × tag 𝐵))
6		df-bj-1upl 32179	. . . . 5 ⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴)
7		bj-pr2eq 32197	. . . . 5 ⊢ (⦅𝐴⦆ = ({∅} × tag 𝐴) → pr2 ⦅𝐴⦆ = pr2 ({∅} × tag 𝐴))
8	6, 7	ax-mp 5	. . . 4 ⊢ pr2 ⦅𝐴⦆ = pr2 ({∅} × tag 𝐴)
9		bj-pr2val 32199	. . . 4 ⊢ pr2 ({∅} × tag 𝐴) = if(∅ = 1𝑜, 𝐴, ∅)
10		1n0 7462	. . . . . 6 ⊢ 1𝑜 ≠ ∅
11	10	nesymi 2839	. . . . 5 ⊢ ¬ ∅ = 1𝑜
12	11	iffalsei 4046	. . . 4 ⊢ if(∅ = 1𝑜, 𝐴, ∅) = ∅
13	8, 9, 12	3eqtri 2636	. . 3 ⊢ pr2 ⦅𝐴⦆ = ∅
14		bj-pr2val 32199	. . . 4 ⊢ pr2 ({1𝑜} × tag 𝐵) = if(1𝑜 = 1𝑜, 𝐵, ∅)
15		eqid 2610	. . . . 5 ⊢ 1𝑜 = 1𝑜
16	15	iftruei 4043	. . . 4 ⊢ if(1𝑜 = 1𝑜, 𝐵, ∅) = 𝐵
17	14, 16	eqtri 2632	. . 3 ⊢ pr2 ({1𝑜} × tag 𝐵) = 𝐵
18	13, 17	uneq12i 3727	. 2 ⊢ (pr2 ⦅𝐴⦆ ∪ pr2 ({1𝑜} × tag 𝐵)) = (∅ ∪ 𝐵)
19		uncom 3719	. . 3 ⊢ (∅ ∪ 𝐵) = (𝐵 ∪ ∅)
20		un0 3919	. . 3 ⊢ (𝐵 ∪ ∅) = 𝐵
21	19, 20	eqtri 2632	. 2 ⊢ (∅ ∪ 𝐵) = 𝐵
22	5, 18, 21	3eqtri 2636	1 ⊢ pr2 ⦅𝐴, 𝐵⦆ = 𝐵

Syntax hints:   = wceq 1475   ∪ cun 3538  ∅c0 3874  ifcif 4036  {csn 4125   × cxp 5036  1_𝑜c1o 7440  tag bj-ctag 32155  ⦅bj-c1upl 32178  ⦅bj-c2uple 32191  pr₂ bj-cpr2 32195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-suc 5646  df-1o 7447  df-bj-sngl 32147  df-bj-tag 32156  df-bj-proj 32172  df-bj-1upl 32179  df-bj-2upl 32192  df-bj-pr2 32196

This theorem is referenced by:  bj-2uplth  32202  bj-2uplex  32203