Theorem bj-pr22val 37675

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 37667	. . . 4 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
2		bj-pr2eq 37672	. . . 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 37673	. . 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 37654	. . . . 5 ⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴)
7		bj-pr2eq 37672	. . . . 5 ⊢ (⦅𝐴⦆ = ({∅} × tag 𝐴) → pr2 ⦅𝐴⦆ = pr2 ({∅} × tag 𝐴))
8	6, 7	ax-mp 5	. . . 4 ⊢ pr2 ⦅𝐴⦆ = pr2 ({∅} × tag 𝐴)
9		bj-pr2val 37674	. . . 4 ⊢ pr2 ({∅} × tag 𝐴) = if(∅ = 1o, 𝐴, ∅)
10		1n0 8468	. . . . . 6 ⊢ 1o ≠ ∅
11	10	nesymi 3015	. . . . 5 ⊢ ¬ ∅ = 1o
12	11	iffalsei 4497	. . . 4 ⊢ if(∅ = 1o, 𝐴, ∅) = ∅
13	8, 9, 12	3eqtri 2790	. . 3 ⊢ pr2 ⦅𝐴⦆ = ∅
14		bj-pr2val 37674	. . . 4 ⊢ pr2 ({1o} × tag 𝐵) = if(1o = 1o, 𝐵, ∅)
15		eqid 2763	. . . . 5 ⊢ 1o = 1o
16	15	iftruei 4494	. . . 4 ⊢ if(1o = 1o, 𝐵, ∅) = 𝐵
17	14, 16	eqtri 2786	. . 3 ⊢ pr2 ({1o} × tag 𝐵) = 𝐵
18	13, 17	uneq12i 4120	. 2 ⊢ (pr2 ⦅𝐴⦆ ∪ pr2 ({1o} × tag 𝐵)) = (∅ ∪ 𝐵)
19		0un 4353	. 2 ⊢ (∅ ∪ 𝐵) = 𝐵
20	5, 18, 19	3eqtri 2790	1 ⊢ pr2 ⦅𝐴, 𝐵⦆ = 𝐵

Syntax hints:   = wceq 1570   ∪ cun 3903  ∅c0 4286  ifcif 4487  {csn 4589   × cxp 5659  1_oc1o 8442  tag bj-ctag 37630  ⦅bj-c1upl 37653  ⦅bj-c2uple 37666  pr₂ bj-cpr2 37670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-xp 5667  df-rel 5668  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-suc 6366  df-1o 8449  df-bj-sngl 37622  df-bj-tag 37631  df-bj-proj 37647  df-bj-1upl 37654  df-bj-2upl 37667  df-bj-pr2 37671

This theorem is referenced by:  bj-2uplth  37677  bj-2uplex  37678