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Theorem bj-pr22val 31569
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
StepHypRef Expression
1 df-bj-2upl 31561 . . . 4  |- (| A,  B|)  =  ((| A|)  u.  ( { 1o }  X. tag  B
) )
2 bj-pr2eq 31566 . . . 4  |-  ((| A,  B|)  =  ((| A|)  u.  ( { 1o }  X. tag  B ) )  -> pr2 (| A,  B|)  = pr2  ((| A|)  u.  ( { 1o }  X. tag  B ) ) )
31, 2ax-mp 5 . . 3  |- pr2 (| A,  B|)  = pr2  ((| A|)  u.  ( { 1o }  X. tag  B
) )
4 bj-pr2un 31567 . . 3  |- pr2  ((| A|)  u.  ( { 1o }  X. tag  B ) )  =  (pr2 (| A|)  u. pr2  ( { 1o }  X. tag  B ) )
53, 4eqtri 2451 . 2  |- pr2 (| A,  B|)  =  (pr2 (| A|)  u. pr2  ( { 1o }  X. tag  B ) )
6 df-bj-1upl 31548 . . . . 5  |- (| A|)  =  ( { (/) }  X. tag  A
)
7 bj-pr2eq 31566 . . . . 5  |-  ((| A|)  =  ( { (/) }  X. tag  A )  -> pr2 (| A|)  = pr2  ( { (/) }  X. tag  A ) )
86, 7ax-mp 5 . . . 4  |- pr2 (| A|)  = pr2  ( { (/) }  X. tag  A
)
9 bj-pr2val 31568 . . . 4  |- pr2  ( { (/)
}  X. tag  A )  =  if ( (/)  =  1o ,  A ,  (/) )
10 1n0 7202 . . . . . 6  |-  1o  =/=  (/)
1110nesymi 2697 . . . . 5  |-  -.  (/)  =  1o
1211iffalsei 3919 . . . 4  |-  if (
(/)  =  1o ,  A ,  (/) )  =  (/)
138, 9, 123eqtri 2455 . . 3  |- pr2 (| A|)  =  (/)
14 bj-pr2val 31568 . . . 4  |- pr2  ( { 1o }  X. tag  B )  =  if ( 1o  =  1o ,  B ,  (/) )
15 eqid 2422 . . . . 5  |-  1o  =  1o
1615iftruei 3916 . . . 4  |-  if ( 1o  =  1o ,  B ,  (/) )  =  B
1714, 16eqtri 2451 . . 3  |- pr2  ( { 1o }  X. tag  B )  =  B
1813, 17uneq12i 3618 . 2  |-  (pr2 (| A|)  u. pr2  ( { 1o }  X. tag  B
) )  =  (
(/)  u.  B )
19 uncom 3610 . . 3  |-  ( (/)  u.  B )  =  ( B  u.  (/) )
20 un0 3787 . . 3  |-  ( B  u.  (/) )  =  B
2119, 20eqtri 2451 . 2  |-  ( (/)  u.  B )  =  B
225, 18, 213eqtri 2455 1  |- pr2 (| A,  B|)  =  B
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437    u. cun 3434   (/)c0 3761   ifcif 3909   {csn 3996    X. cxp 4848   1oc1o 7180  tag bj-ctag 31524  (|bj-c1upl 31547  (|bj-c2uple 31560  pr2 bj-cpr2 31564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pr 4657  ax-un 6594
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-xp 4856  df-rel 4857  df-cnv 4858  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-suc 5445  df-1o 7187  df-bj-sngl 31516  df-bj-tag 31525  df-bj-proj 31541  df-bj-1upl 31548  df-bj-2upl 31561  df-bj-pr2 31565
This theorem is referenced by:  bj-2uplth  31571  bj-2uplex  31572
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