Theorem bj-pr22val 34333

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 34325	. . . 4 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
2		bj-pr2eq 34330	. . . 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 34331	. . 3 ⊢ pr2 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) = (pr2 ⦅𝐴⦆ ∪ pr2 ({1o} × tag 𝐵))
5	3, 4	eqtri 2846	. 2 ⊢ pr2 ⦅𝐴, 𝐵⦆ = (pr2 ⦅𝐴⦆ ∪ pr2 ({1o} × tag 𝐵))
6		df-bj-1upl 34312	. . . . 5 ⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴)
7		bj-pr2eq 34330	. . . . 5 ⊢ (⦅𝐴⦆ = ({∅} × tag 𝐴) → pr2 ⦅𝐴⦆ = pr2 ({∅} × tag 𝐴))
8	6, 7	ax-mp 5	. . . 4 ⊢ pr2 ⦅𝐴⦆ = pr2 ({∅} × tag 𝐴)
9		bj-pr2val 34332	. . . 4 ⊢ pr2 ({∅} × tag 𝐴) = if(∅ = 1o, 𝐴, ∅)
10		1n0 8121	. . . . . 6 ⊢ 1o ≠ ∅
11	10	nesymi 3075	. . . . 5 ⊢ ¬ ∅ = 1o
12	11	iffalsei 4479	. . . 4 ⊢ if(∅ = 1o, 𝐴, ∅) = ∅
13	8, 9, 12	3eqtri 2850	. . 3 ⊢ pr2 ⦅𝐴⦆ = ∅
14		bj-pr2val 34332	. . . 4 ⊢ pr2 ({1o} × tag 𝐵) = if(1o = 1o, 𝐵, ∅)
15		eqid 2823	. . . . 5 ⊢ 1o = 1o
16	15	iftruei 4476	. . . 4 ⊢ if(1o = 1o, 𝐵, ∅) = 𝐵
17	14, 16	eqtri 2846	. . 3 ⊢ pr2 ({1o} × tag 𝐵) = 𝐵
18	13, 17	uneq12i 4139	. 2 ⊢ (pr2 ⦅𝐴⦆ ∪ pr2 ({1o} × tag 𝐵)) = (∅ ∪ 𝐵)
19		0un 4348	. 2 ⊢ (∅ ∪ 𝐵) = 𝐵
20	5, 18, 19	3eqtri 2850	1 ⊢ pr2 ⦅𝐴, 𝐵⦆ = 𝐵

Syntax hints:   = wceq 1537   ∪ cun 3936  ∅c0 4293  ifcif 4469  {csn 4569   × cxp 5555  1_oc1o 8097  tag bj-ctag 34288  ⦅bj-c1upl 34311  ⦅bj-c2uple 34324  pr₂ bj-cpr2 34328

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-tr 5175  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-ord 6196  df-on 6197  df-suc 6199  df-1o 8104  df-bj-sngl 34280  df-bj-tag 34289  df-bj-proj 34305  df-bj-1upl 34312  df-bj-2upl 34325  df-bj-pr2 34329

This theorem is referenced by:  bj-2uplth  34335  bj-2uplex  34336