Theorem trtxp 5782

Description: Trinary relationship over a tail cross product. (Contributed by SF, 13-Feb-2015.)

Assertion

Ref	Expression
trtxp	⊢ (A(R ⊗ S)⟨B, C⟩ ↔ (ARB ∧ ASC))

Proof of Theorem trtxp

Dummy variables x y z t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brex 4690	. . 3 ⊢ (A(R ⊗ S)⟨B, C⟩ → (A ∈ V ∧ ⟨B, C⟩ ∈ V))
2		opexb 4604	. . . 4 ⊢ (⟨B, C⟩ ∈ V ↔ (B ∈ V ∧ C ∈ V))
3	2	anbi2i 675	. . 3 ⊢ ((A ∈ V ∧ ⟨B, C⟩ ∈ V) ↔ (A ∈ V ∧ (B ∈ V ∧ C ∈ V)))
4	1, 3	sylib 188	. 2 ⊢ (A(R ⊗ S)⟨B, C⟩ → (A ∈ V ∧ (B ∈ V ∧ C ∈ V)))
5		brex 4690	. . . 4 ⊢ (ARB → (A ∈ V ∧ B ∈ V))
6		brex 4690	. . . 4 ⊢ (ASC → (A ∈ V ∧ C ∈ V))
7	5, 6	anim12i 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)))
9	7, 8	sylibr 203	. 2 ⊢ ((ARB ∧ ASC) → (A ∈ V ∧ (B ∈ V ∧ C ∈ V)))
10		breq1 4643	. . . . . 6 ⊢ (x = A → (x(R ⊗ S)⟨B, C⟩ ↔ A(R ⊗ S)⟨B, C⟩))
11		breq1 4643	. . . . . . 7 ⊢ (x = A → (xRB ↔ ARB))
12		breq1 4643	. . . . . . 7 ⊢ (x = A → (xSC ↔ ASC))
13	11, 12	anbi12d 691	. . . . . 6 ⊢ (x = A → ((xRB ∧ xSC) ↔ (ARB ∧ ASC)))
14	10, 13	bibi12d 312	. . . . 5 ⊢ (x = A → ((x(R ⊗ S)⟨B, C⟩ ↔ (xRB ∧ xSC)) ↔ (A(R ⊗ S)⟨B, C⟩ ↔ (ARB ∧ ASC))))
15	14	imbi2d 307	. . . 4 ⊢ (x = A → (((B ∈ V ∧ C ∈ V) → (x(R ⊗ S)⟨B, C⟩ ↔ (xRB ∧ xSC))) ↔ ((B ∈ V ∧ C ∈ V) → (A(R ⊗ S)⟨B, C⟩ ↔ (ARB ∧ ASC)))))
16		opeq1 4579	. . . . . . 7 ⊢ (y = B → ⟨y, z⟩ = ⟨B, z⟩)
17	16	breq2d 4652	. . . . . 6 ⊢ (y = B → (x(R ⊗ S)⟨y, z⟩ ↔ x(R ⊗ S)⟨B, z⟩))
18		breq2 4644	. . . . . . 7 ⊢ (y = B → (xRy ↔ xRB))
19	18	anbi1d 685	. . . . . 6 ⊢ (y = B → ((xRy ∧ xSz) ↔ (xRB ∧ xSz)))
20	17, 19	bibi12d 312	. . . . 5 ⊢ (y = B → ((x(R ⊗ S)⟨y, z⟩ ↔ (xRy ∧ xSz)) ↔ (x(R ⊗ S)⟨B, z⟩ ↔ (xRB ∧ xSz))))
21		opeq2 4580	. . . . . . 7 ⊢ (z = C → ⟨B, z⟩ = ⟨B, C⟩)
22	21	breq2d 4652	. . . . . 6 ⊢ (z = C → (x(R ⊗ S)⟨B, z⟩ ↔ x(R ⊗ S)⟨B, C⟩))
23		breq2 4644	. . . . . . 7 ⊢ (z = C → (xSz ↔ xSC))
24	23	anbi2d 684	. . . . . 6 ⊢ (z = C → ((xRB ∧ xSz) ↔ (xRB ∧ xSC)))
25	22, 24	bibi12d 312	. . . . 5 ⊢ (z = C → ((x(R ⊗ S)⟨B, z⟩ ↔ (xRB ∧ xSz)) ↔ (x(R ⊗ S)⟨B, C⟩ ↔ (xRB ∧ xSC))))
26		df-txp 5737	. . . . . . 7 ⊢ (R ⊗ S) = ((◡1st ∘ R) ∩ (◡2nd ∘ S))
27	26	breqi 4646	. . . . . 6 ⊢ (x(R ⊗ S)⟨y, z⟩ ↔ x((◡1st ∘ R) ∩ (◡2nd ∘ S))⟨y, z⟩)
28		brin 4694	. . . . . 6 ⊢ (x((◡1st ∘ R) ∩ (◡2nd ∘ S))⟨y, z⟩ ↔ (x(◡1st ∘ R)⟨y, z⟩ ∧ x(◡2nd ∘ S)⟨y, z⟩))
29		brco 4884	. . . . . . . 8 ⊢ (x(◡1st ∘ R)⟨y, z⟩ ↔ ∃t(xRt ∧ t◡1st ⟨y, z⟩))
30		ancom 437	. . . . . . . . . 10 ⊢ ((xRt ∧ t◡1st ⟨y, z⟩) ↔ (t◡1st ⟨y, z⟩ ∧ xRt))
31		brcnv 4893	. . . . . . . . . . . 12 ⊢ (t◡1st ⟨y, z⟩ ↔ ⟨y, z⟩1st t)
32		vex 2863	. . . . . . . . . . . . 13 ⊢ y ∈ V
33		vex 2863	. . . . . . . . . . . . 13 ⊢ z ∈ V
34	32, 33	opbr1st 5502	. . . . . . . . . . . 12 ⊢ (⟨y, z⟩1st t ↔ y = t)
35		equcom 1680	. . . . . . . . . . . 12 ⊢ (y = t ↔ t = y)
36	31, 34, 35	3bitri 262	. . . . . . . . . . 11 ⊢ (t◡1st ⟨y, z⟩ ↔ t = y)
37	36	anbi1i 676	. . . . . . . . . 10 ⊢ ((t◡1st ⟨y, z⟩ ∧ xRt) ↔ (t = y ∧ xRt))
38	30, 37	bitri 240	. . . . . . . . 9 ⊢ ((xRt ∧ t◡1st ⟨y, z⟩) ↔ (t = y ∧ xRt))
39	38	exbii 1582	. . . . . . . 8 ⊢ (∃t(xRt ∧ t◡1st ⟨y, z⟩) ↔ ∃t(t = y ∧ xRt))
40		breq2 4644	. . . . . . . . 9 ⊢ (t = y → (xRt ↔ xRy))
41	32, 40	ceqsexv 2895	. . . . . . . 8 ⊢ (∃t(t = y ∧ xRt) ↔ xRy)
42	29, 39, 41	3bitri 262	. . . . . . 7 ⊢ (x(◡1st ∘ R)⟨y, z⟩ ↔ xRy)
43		brco 4884	. . . . . . . 8 ⊢ (x(◡2nd ∘ S)⟨y, z⟩ ↔ ∃t(xSt ∧ t◡2nd ⟨y, z⟩))
44		ancom 437	. . . . . . . . . 10 ⊢ ((xSt ∧ t◡2nd ⟨y, z⟩) ↔ (t◡2nd ⟨y, z⟩ ∧ xSt))
45		brcnv 4893	. . . . . . . . . . . 12 ⊢ (t◡2nd ⟨y, z⟩ ↔ ⟨y, z⟩2nd t)
46	32, 33	opbr2nd 5503	. . . . . . . . . . . 12 ⊢ (⟨y, z⟩2nd t ↔ z = t)
47		equcom 1680	. . . . . . . . . . . 12 ⊢ (z = t ↔ t = z)
48	45, 46, 47	3bitri 262	. . . . . . . . . . 11 ⊢ (t◡2nd ⟨y, z⟩ ↔ t = z)
49	48	anbi1i 676	. . . . . . . . . 10 ⊢ ((t◡2nd ⟨y, z⟩ ∧ xSt) ↔ (t = z ∧ xSt))
50	44, 49	bitri 240	. . . . . . . . 9 ⊢ ((xSt ∧ t◡2nd ⟨y, z⟩) ↔ (t = z ∧ xSt))
51	50	exbii 1582	. . . . . . . 8 ⊢ (∃t(xSt ∧ t◡2nd ⟨y, z⟩) ↔ ∃t(t = z ∧ xSt))
52		breq2 4644	. . . . . . . . 9 ⊢ (t = z → (xSt ↔ xSz))
53	33, 52	ceqsexv 2895	. . . . . . . 8 ⊢ (∃t(t = z ∧ xSt) ↔ xSz)
54	43, 51, 53	3bitri 262	. . . . . . 7 ⊢ (x(◡2nd ∘ S)⟨y, z⟩ ↔ xSz)
55	42, 54	anbi12i 678	. . . . . 6 ⊢ ((x(◡1st ∘ R)⟨y, z⟩ ∧ x(◡2nd ∘ S)⟨y, z⟩) ↔ (xRy ∧ xSz))
56	27, 28, 55	3bitri 262	. . . . 5 ⊢ (x(R ⊗ S)⟨y, z⟩ ↔ (xRy ∧ xSz))
57	20, 25, 56	vtocl2g 2919	. . . 4 ⊢ ((B ∈ V ∧ C ∈ V) → (x(R ⊗ S)⟨B, C⟩ ↔ (xRB ∧ xSC)))
58	15, 57	vtoclg 2915	. . 3 ⊢ (A ∈ V → ((B ∈ V ∧ C ∈ V) → (A(R ⊗ S)⟨B, C⟩ ↔ (ARB ∧ ASC))))
59	58	imp 418	. 2 ⊢ ((A ∈ V ∧ (B ∈ V ∧ C ∈ V)) → (A(R ⊗ S)⟨B, C⟩ ↔ (ARB ∧ ASC)))
60	4, 9, 59	pm5.21nii 342	1 ⊢ (A(R ⊗ S)⟨B, C⟩ ↔ (ARB ∧ ASC))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ∩ cin 3209  ⟨cop 4562   class class class wbr 4640  1^st c1st 4718   ∘ ccom 4722  ^◡ccnv 4772  2^nd c2nd 4784   ⊗ ctxp 5736

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-co 4727  df-cnv 4786  df-2nd 4798  df-txp 5737

This theorem is referenced by:  oteltxp  5783  txpcofun  5804  addcfnex  5825  qrpprod  5837  xpassenlem  6057  xpassen  6058  enmap2lem1  6064  enmap1lem1  6070  ovmuc  6131  ceex  6175  nncdiv3lem1  6276  nchoicelem10  6299