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Theorem bj-pr22val 34319
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 34311 . . . 4 𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
2 bj-pr2eq 34316 . . . 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 34317 . . 3 pr2 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) = (pr2𝐴⦆ ∪ pr2 ({1o} × tag 𝐵))
53, 4eqtri 2842 . 2 pr2𝐴, 𝐵⦆ = (pr2𝐴⦆ ∪ pr2 ({1o} × tag 𝐵))
6 df-bj-1upl 34298 . . . . 5 𝐴⦆ = ({∅} × tag 𝐴)
7 bj-pr2eq 34316 . . . . 5 (⦅𝐴⦆ = ({∅} × tag 𝐴) → pr2𝐴⦆ = pr2 ({∅} × tag 𝐴))
86, 7ax-mp 5 . . . 4 pr2𝐴⦆ = pr2 ({∅} × tag 𝐴)
9 bj-pr2val 34318 . . . 4 pr2 ({∅} × tag 𝐴) = if(∅ = 1o, 𝐴, ∅)
10 1n0 8111 . . . . . 6 1o ≠ ∅
1110nesymi 3071 . . . . 5 ¬ ∅ = 1o
1211iffalsei 4475 . . . 4 if(∅ = 1o, 𝐴, ∅) = ∅
138, 9, 123eqtri 2846 . . 3 pr2𝐴⦆ = ∅
14 bj-pr2val 34318 . . . 4 pr2 ({1o} × tag 𝐵) = if(1o = 1o, 𝐵, ∅)
15 eqid 2819 . . . . 5 1o = 1o
1615iftruei 4472 . . . 4 if(1o = 1o, 𝐵, ∅) = 𝐵
1714, 16eqtri 2842 . . 3 pr2 ({1o} × tag 𝐵) = 𝐵
1813, 17uneq12i 4135 . 2 (pr2𝐴⦆ ∪ pr2 ({1o} × tag 𝐵)) = (∅ ∪ 𝐵)
19 0un 4344 . 2 (∅ ∪ 𝐵) = 𝐵
205, 18, 193eqtri 2846 1 pr2𝐴, 𝐵⦆ = 𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1530  cun 3932  c0 4289  ifcif 4465  {csn 4559   × cxp 5546  1oc1o 8087  tag bj-ctag 34274  bj-c1upl 34297  bj-c2uple 34310  pr2 bj-cpr2 34314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-suc 6190  df-1o 8094  df-bj-sngl 34266  df-bj-tag 34275  df-bj-proj 34291  df-bj-1upl 34298  df-bj-2upl 34311  df-bj-pr2 34315
This theorem is referenced by:  bj-2uplth  34321  bj-2uplex  34322
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