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Theorem bj-pr22val 32200
 Description: Value of the second projection of a couple. (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-pr22val pr2𝐴, 𝐵⦆ = 𝐵

Proof of Theorem bj-pr22val
StepHypRef Expression
1 df-bj-2upl 32192 . . . 4 𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐵))
2 bj-pr2eq 32197 . . . 4 (⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐵)) → pr2𝐴, 𝐵⦆ = pr2 (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐵)))
31, 2ax-mp 5 . . 3 pr2𝐴, 𝐵⦆ = pr2 (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐵))
4 bj-pr2un 32198 . . 3 pr2 (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐵)) = (pr2𝐴⦆ ∪ pr2 ({1𝑜} × tag 𝐵))
53, 4eqtri 2632 . 2 pr2𝐴, 𝐵⦆ = (pr2𝐴⦆ ∪ pr2 ({1𝑜} × tag 𝐵))
6 df-bj-1upl 32179 . . . . 5 𝐴⦆ = ({∅} × tag 𝐴)
7 bj-pr2eq 32197 . . . . 5 (⦅𝐴⦆ = ({∅} × tag 𝐴) → pr2𝐴⦆ = pr2 ({∅} × tag 𝐴))
86, 7ax-mp 5 . . . 4 pr2𝐴⦆ = pr2 ({∅} × tag 𝐴)
9 bj-pr2val 32199 . . . 4 pr2 ({∅} × tag 𝐴) = if(∅ = 1𝑜, 𝐴, ∅)
10 1n0 7462 . . . . . 6 1𝑜 ≠ ∅
1110nesymi 2839 . . . . 5 ¬ ∅ = 1𝑜
1211iffalsei 4046 . . . 4 if(∅ = 1𝑜, 𝐴, ∅) = ∅
138, 9, 123eqtri 2636 . . 3 pr2𝐴⦆ = ∅
14 bj-pr2val 32199 . . . 4 pr2 ({1𝑜} × tag 𝐵) = if(1𝑜 = 1𝑜, 𝐵, ∅)
15 eqid 2610 . . . . 5 1𝑜 = 1𝑜
1615iftruei 4043 . . . 4 if(1𝑜 = 1𝑜, 𝐵, ∅) = 𝐵
1714, 16eqtri 2632 . . 3 pr2 ({1𝑜} × tag 𝐵) = 𝐵
1813, 17uneq12i 3727 . 2 (pr2𝐴⦆ ∪ pr2 ({1𝑜} × tag 𝐵)) = (∅ ∪ 𝐵)
19 uncom 3719 . . 3 (∅ ∪ 𝐵) = (𝐵 ∪ ∅)
20 un0 3919 . . 3 (𝐵 ∪ ∅) = 𝐵
2119, 20eqtri 2632 . 2 (∅ ∪ 𝐵) = 𝐵
225, 18, 213eqtri 2636 1 pr2𝐴, 𝐵⦆ = 𝐵
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∪ cun 3538  ∅c0 3874  ifcif 4036  {csn 4125   × cxp 5036  1𝑜c1o 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
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