Theorem bj-pr22val 38147

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 38139	. . . 4 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
2		bj-pr2eq 38144	. . . 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 38145	. . 3 ⊢ pr2 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) = (pr2 ⦅𝐴⦆ ∪ pr2 ({1o} × tag 𝐵))
5	3, 4	eqtri 2786	. 2 ⊢ pr2 ⦅𝐴, 𝐵⦆ = (pr2 ⦅𝐴⦆ ∪ pr2 ({1o} × tag 𝐵))
6		df-bj-1upl 38126	. . . . 5 ⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴)
7		bj-pr2eq 38144	. . . . 5 ⊢ (⦅𝐴⦆ = ({∅} × tag 𝐴) → pr2 ⦅𝐴⦆ = pr2 ({∅} × tag 𝐴))
8	6, 7	ax-mp 5	. . . 4 ⊢ pr2 ⦅𝐴⦆ = pr2 ({∅} × tag 𝐴)
9		bj-pr2val 38146	. . . 4 ⊢ pr2 ({∅} × tag 𝐴) = if(∅ = 1o, 𝐴, ∅)
10		1n0 8533	. . . . . 6 ⊢ 1o ≠ ∅
11	10	nesymi 3016	. . . . 5 ⊢ ¬ ∅ = 1o
12	11	iffalsei 4495	. . . 4 ⊢ if(∅ = 1o, 𝐴, ∅) = ∅
13	8, 9, 12	3eqtri 2790	. . 3 ⊢ pr2 ⦅𝐴⦆ = ∅
14		bj-pr2val 38146	. . . 4 ⊢ pr2 ({1o} × tag 𝐵) = if(1o = 1o, 𝐵, ∅)
15		eqid 2763	. . . . 5 ⊢ 1o = 1o
16	15	iftruei 4492	. . . 4 ⊢ if(1o = 1o, 𝐵, ∅) = 𝐵
17	14, 16	eqtri 2786	. . 3 ⊢ pr2 ({1o} × tag 𝐵) = 𝐵
18	13, 17	uneq12i 4168	. 2 ⊢ (pr2 ⦅𝐴⦆ ∪ pr2 ({1o} × tag 𝐵)) = (∅ ∪ 𝐵)
19		0un 4349	. 2 ⊢ (∅ ∪ 𝐵) = 𝐵
20	5, 18, 19	3eqtri 2790	1 ⊢ pr2 ⦅𝐴, 𝐵⦆ = 𝐵

This proof depends on syntax axioms:   = wceq 1570   ∪ cun 4152  ∅c0 4281  ifcif 4485  {csn 4588   × cxp 5656  1_oc1o 8507  tag bj-ctag 38102  ⦅bj-c1upl 38125  ⦅bj-c2uple 38138  pr₂ bj-cpr2 38142

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-xp 5664  df-rel 5665  df-cnv 5666  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-suc 6394  df-1o 8514  df-bj-sngl 38094  df-bj-tag 38103  df-bj-proj 38119  df-bj-1upl 38126  df-bj-2upl 38139  df-bj-pr2 38143

This theorem is used by:  bj-2uplth  38149  bj-2uplex  38150