Theorem bj-pr22val 38059

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 38051	. . . 4 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
2		bj-pr2eq 38056	. . . 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 38057	. . 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 38038	. . . . 5 ⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴)
7		bj-pr2eq 38056	. . . . 5 ⊢ (⦅𝐴⦆ = ({∅} × tag 𝐴) → pr2 ⦅𝐴⦆ = pr2 ({∅} × tag 𝐴))
8	6, 7	ax-mp 5	. . . 4 ⊢ pr2 ⦅𝐴⦆ = pr2 ({∅} × tag 𝐴)
9		bj-pr2val 38058	. . . 4 ⊢ pr2 ({∅} × tag 𝐴) = if(∅ = 1o, 𝐴, ∅)
10		1n0 8506	. . . . . 6 ⊢ 1o ≠ ∅
11	10	nesymi 3015	. . . . 5 ⊢ ¬ ∅ = 1o
12	11	iffalsei 4494	. . . 4 ⊢ if(∅ = 1o, 𝐴, ∅) = ∅
13	8, 9, 12	3eqtri 2790	. . 3 ⊢ pr2 ⦅𝐴⦆ = ∅
14		bj-pr2val 38058	. . . 4 ⊢ pr2 ({1o} × tag 𝐵) = if(1o = 1o, 𝐵, ∅)
15		eqid 2763	. . . . 5 ⊢ 1o = 1o
16	15	iftruei 4491	. . . 4 ⊢ if(1o = 1o, 𝐵, ∅) = 𝐵
17	14, 16	eqtri 2786	. . 3 ⊢ pr2 ({1o} × tag 𝐵) = 𝐵
18	13, 17	uneq12i 4167	. 2 ⊢ (pr2 ⦅𝐴⦆ ∪ pr2 ({1o} × tag 𝐵)) = (∅ ∪ 𝐵)
19		0un 4348	. 2 ⊢ (∅ ∪ 𝐵) = 𝐵
20	5, 18, 19	3eqtri 2790	1 ⊢ pr2 ⦅𝐴, 𝐵⦆ = 𝐵

This proof depends on syntax axioms:   = wceq 1569   ∪ cun 4151  ∅c0 4280  ifcif 4484  {csn 4587   × cxp 5652  1_oc1o 8480  tag bj-ctag 38014  ⦅bj-c1upl 38037  ⦅bj-c2uple 38050  pr₂ bj-cpr2 38054

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-br 5106  df-opab 5170  df-xp 5660  df-rel 5661  df-cnv 5662  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-suc 6374  df-1o 8487  df-bj-sngl 38006  df-bj-tag 38015  df-bj-proj 38031  df-bj-1upl 38038  df-bj-2upl 38051  df-bj-pr2 38055

This theorem is used by:  bj-2uplth  38061  bj-2uplex  38062