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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-1 5  ax-2 6  ax-3 7  ax-mp 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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