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Theorem trtxp 5781
Description: Trinary relationship over a tail cross product. (Contributed by SF, 13-Feb-2015.)
Assertion
Ref Expression
trtxp (A(RS)B, C ↔ (ARB ASC))

Proof of Theorem trtxp
Dummy variables x y z t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brex 4689 . . 3 (A(RS)B, C → (A V B, C V))
2 opexb 4603 . . . 4 (B, C V ↔ (B V C V))
32anbi2i 675 . . 3 ((A V B, C V) ↔ (A V (B V C V)))
41, 3sylib 188 . 2 (A(RS)B, C → (A V (B V C V)))
5 brex 4689 . . . 4 (ARB → (A V B V))
6 brex 4689 . . . 4 (ASC → (A V C V))
75, 6anim12i 549 . . 3 ((ARB ASC) → ((A V B V) (A V C V)))
8 anandi 801 . . 3 ((A V (B V C V)) ↔ ((A V B V) (A V C V)))
97, 8sylibr 203 . 2 ((ARB ASC) → (A V (B V C V)))
10 breq1 4642 . . . . . 6 (x = A → (x(RS)B, CA(RS)B, C))
11 breq1 4642 . . . . . . 7 (x = A → (xRBARB))
12 breq1 4642 . . . . . . 7 (x = A → (xSCASC))
1311, 12anbi12d 691 . . . . . 6 (x = A → ((xRB xSC) ↔ (ARB ASC)))
1410, 13bibi12d 312 . . . . 5 (x = A → ((x(RS)B, C ↔ (xRB xSC)) ↔ (A(RS)B, C ↔ (ARB ASC))))
1514imbi2d 307 . . . 4 (x = A → (((B V C V) → (x(RS)B, C ↔ (xRB xSC))) ↔ ((B V C V) → (A(RS)B, C ↔ (ARB ASC)))))
16 opeq1 4578 . . . . . . 7 (y = By, z = B, z)
1716breq2d 4651 . . . . . 6 (y = B → (x(RS)y, zx(RS)B, z))
18 breq2 4643 . . . . . . 7 (y = B → (xRyxRB))
1918anbi1d 685 . . . . . 6 (y = B → ((xRy xSz) ↔ (xRB xSz)))
2017, 19bibi12d 312 . . . . 5 (y = B → ((x(RS)y, z ↔ (xRy xSz)) ↔ (x(RS)B, z ↔ (xRB xSz))))
21 opeq2 4579 . . . . . . 7 (z = CB, z = B, C)
2221breq2d 4651 . . . . . 6 (z = C → (x(RS)B, zx(RS)B, C))
23 breq2 4643 . . . . . . 7 (z = C → (xSzxSC))
2423anbi2d 684 . . . . . 6 (z = C → ((xRB xSz) ↔ (xRB xSC)))
2522, 24bibi12d 312 . . . . 5 (z = C → ((x(RS)B, z ↔ (xRB xSz)) ↔ (x(RS)B, C ↔ (xRB xSC))))
26 df-txp 5736 . . . . . . 7 (RS) = ((1st R) ∩ (2nd S))
2726breqi 4645 . . . . . 6 (x(RS)y, zx((1st R) ∩ (2nd S))y, z)
28 brin 4693 . . . . . 6 (x((1st R) ∩ (2nd S))y, z ↔ (x(1st R)y, z x(2nd S)y, z))
29 brco 4883 . . . . . . . 8 (x(1st R)y, zt(xRt t1st y, z))
30 ancom 437 . . . . . . . . . 10 ((xRt t1st y, z) ↔ (t1st y, z xRt))
31 brcnv 4892 . . . . . . . . . . . 12 (t1st y, zy, z1st t)
32 vex 2862 . . . . . . . . . . . . 13 y V
33 vex 2862 . . . . . . . . . . . . 13 z V
3432, 33opbr1st 5501 . . . . . . . . . . . 12 (y, z1st ty = t)
35 equcom 1680 . . . . . . . . . . . 12 (y = tt = y)
3631, 34, 353bitri 262 . . . . . . . . . . 11 (t1st y, zt = y)
3736anbi1i 676 . . . . . . . . . 10 ((t1st y, z xRt) ↔ (t = y xRt))
3830, 37bitri 240 . . . . . . . . 9 ((xRt t1st y, z) ↔ (t = y xRt))
3938exbii 1582 . . . . . . . 8 (t(xRt t1st y, z) ↔ t(t = y xRt))
40 breq2 4643 . . . . . . . . 9 (t = y → (xRtxRy))
4132, 40ceqsexv 2894 . . . . . . . 8 (t(t = y xRt) ↔ xRy)
4229, 39, 413bitri 262 . . . . . . 7 (x(1st R)y, zxRy)
43 brco 4883 . . . . . . . 8 (x(2nd S)y, zt(xSt t2nd y, z))
44 ancom 437 . . . . . . . . . 10 ((xSt t2nd y, z) ↔ (t2nd y, z xSt))
45 brcnv 4892 . . . . . . . . . . . 12 (t2nd y, zy, z2nd t)
4632, 33opbr2nd 5502 . . . . . . . . . . . 12 (y, z2nd tz = t)
47 equcom 1680 . . . . . . . . . . . 12 (z = tt = z)
4845, 46, 473bitri 262 . . . . . . . . . . 11 (t2nd y, zt = z)
4948anbi1i 676 . . . . . . . . . 10 ((t2nd y, z xSt) ↔ (t = z xSt))
5044, 49bitri 240 . . . . . . . . 9 ((xSt t2nd y, z) ↔ (t = z xSt))
5150exbii 1582 . . . . . . . 8 (t(xSt t2nd y, z) ↔ t(t = z xSt))
52 breq2 4643 . . . . . . . . 9 (t = z → (xStxSz))
5333, 52ceqsexv 2894 . . . . . . . 8 (t(t = z xSt) ↔ xSz)
5443, 51, 533bitri 262 . . . . . . 7 (x(2nd S)y, zxSz)
5542, 54anbi12i 678 . . . . . 6 ((x(1st R)y, z x(2nd S)y, z) ↔ (xRy xSz))
5627, 28, 553bitri 262 . . . . 5 (x(RS)y, z ↔ (xRy xSz))
5720, 25, 56vtocl2g 2918 . . . 4 ((B V C V) → (x(RS)B, C ↔ (xRB xSC)))
5815, 57vtoclg 2914 . . 3 (A V → ((B V C V) → (A(RS)B, C ↔ (ARB ASC))))
5958imp 418 . 2 ((A V (B V C V)) → (A(RS)B, C ↔ (ARB ASC)))
604, 9, 59pm5.21nii 342 1 (A(RS)B, C ↔ (ARB ASC))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 176   wa 358  wex 1541   = wceq 1642   wcel 1710  Vcvv 2859  cin 3208  cop 4561   class class class wbr 4639  1st c1st 4717   ccom 4721  ccnv 4771  2nd c2nd 4783  ctxp 5735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-co 4726  df-cnv 4785  df-2nd 4797  df-txp 5736
This theorem is referenced by:  oteltxp  5782  txpcofun  5803  addcfnex  5824  qrpprod  5836  xpassenlem  6056  xpassen  6057  enmap2lem1  6063  enmap1lem1  6069  ovmuc  6130  ceex  6174  nncdiv3lem1  6275  nchoicelem10  6298
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