Theorem trtxp 5782

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

Assertion

Ref	Expression
trtxp	|- (A(R (x) 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 (x) S)<.B, C>. -> (A e. _V /\ <.B, C>. e. _V))
2		opexb 4604	. . . 4 |- (<.B, C>. e. _V <-> (B e. _V /\ C e. _V))
3	2	anbi2i 675	. . 3 |- ((A e. _V /\ <.B, C>. e. _V) <-> (A e. _V /\ (B e. _V /\ C e. _V)))
4	1, 3	sylib 188	. 2 |- (A(R (x) S)<.B, C>. -> (A e. _V /\ (B e. _V /\ C e. _V)))
5		brex 4690	. . . 4 |- (ARB -> (A e. _V /\ B e. _V))
6		brex 4690	. . . 4 |- (ASC -> (A e. _V /\ C e. _V))
7	5, 6	anim12i 549	. . 3 |- ((ARB /\ ASC) -> ((A e. _V /\ B e. _V) /\ (A e. _V /\ C e. _V)))
8		anandi 801	. . 3 |- ((A e. _V /\ (B e. _V /\ C e. _V)) <-> ((A e. _V /\ B e. _V) /\ (A e. _V /\ C e. _V)))
9	7, 8	sylibr 203	. 2 |- ((ARB /\ ASC) -> (A e. _V /\ (B e. _V /\ C e. _V)))
10		breq1 4643	. . . . . 6 |- (x = A -> (x(R (x) S)<.B, C>. <-> A(R (x) 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 (x) S)<.B, C>. <-> (xRB /\ xSC)) <-> (A(R (x) S)<.B, C>. <-> (ARB /\ ASC))))
15	14	imbi2d 307	. . . 4 |- (x = A -> (((B e. _V /\ C e. _V) -> (x(R (x) S)<.B, C>. <-> (xRB /\ xSC))) <-> ((B e. _V /\ C e. _V) -> (A(R (x) S)<.B, C>. <-> (ARB /\ ASC)))))
16		opeq1 4579	. . . . . . 7 |- (y = B -> <.y, z>. = <.B, z>.)
17	16	breq2d 4652	. . . . . 6 |- (y = B -> (x(R (x) S)<.y, z>. <-> x(R (x) 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 (x) S)<.y, z>. <-> (xRy /\ xSz)) <-> (x(R (x) S)<.B, z>. <-> (xRB /\ xSz))))
21		opeq2 4580	. . . . . . 7 |- (z = C -> <.B, z>. = <.B, C>.)
22	21	breq2d 4652	. . . . . 6 |- (z = C -> (x(R (x) S)<.B, z>. <-> x(R (x) 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 (x) S)<.B, z>. <-> (xRB /\ xSz)) <-> (x(R (x) S)<.B, C>. <-> (xRB /\ xSC))))
26		df-txp 5737	. . . . . . 7 |- (R (x) S) = ((`'1st o. R) i^i (`'2nd o. S))
27	26	breqi 4646	. . . . . 6 |- (x(R (x) S)<.y, z>. <-> x((`'1st o. R) i^i (`'2nd o. S))<.y, z>.)
28		brin 4694	. . . . . 6 |- (x((`'1st o. R) i^i (`'2nd o. S))<.y, z>. <-> (x(`'1st o. R)<.y, z>. /\ x(`'2nd o. S)<.y, z>.))
29		brco 4884	. . . . . . . 8 |- (x(`'1st o. R)<.y, z>. <-> E.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>.1stt)
32		vex 2863	. . . . . . . . . . . . 13 |- y e. _V
33		vex 2863	. . . . . . . . . . . . 13 |- z e. _V
34	32, 33	opbr1st 5502	. . . . . . . . . . . 12 |- (<.y, z>.1stt <-> 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 |- (E.t(xRt /\ t`'1st<.y, z>.) <-> E.t(t = y /\ xRt))
40		breq2 4644	. . . . . . . . 9 |- (t = y -> (xRt <-> xRy))
41	32, 40	ceqsexv 2895	. . . . . . . 8 |- (E.t(t = y /\ xRt) <-> xRy)
42	29, 39, 41	3bitri 262	. . . . . . 7 |- (x(`'1st o. R)<.y, z>. <-> xRy)
43		brco 4884	. . . . . . . 8 |- (x(`'2nd o. S)<.y, z>. <-> E.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>.2ndt)
46	32, 33	opbr2nd 5503	. . . . . . . . . . . 12 |- (<.y, z>.2ndt <-> 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 |- (E.t(xSt /\ t`'2nd<.y, z>.) <-> E.t(t = z /\ xSt))
52		breq2 4644	. . . . . . . . 9 |- (t = z -> (xSt <-> xSz))
53	33, 52	ceqsexv 2895	. . . . . . . 8 |- (E.t(t = z /\ xSt) <-> xSz)
54	43, 51, 53	3bitri 262	. . . . . . 7 |- (x(`'2nd o. S)<.y, z>. <-> xSz)
55	42, 54	anbi12i 678	. . . . . 6 |- ((x(`'1st o. R)<.y, z>. /\ x(`'2nd o. S)<.y, z>.) <-> (xRy /\ xSz))
56	27, 28, 55	3bitri 262	. . . . 5 |- (x(R (x) S)<.y, z>. <-> (xRy /\ xSz))
57	20, 25, 56	vtocl2g 2919	. . . 4 |- ((B e. _V /\ C e. _V) -> (x(R (x) S)<.B, C>. <-> (xRB /\ xSC)))
58	15, 57	vtoclg 2915	. . 3 |- (A e. _V -> ((B e. _V /\ C e. _V) -> (A(R (x) S)<.B, C>. <-> (ARB /\ ASC))))
59	58	imp 418	. 2 |- ((A e. _V /\ (B e. _V /\ C e. _V)) -> (A(R (x) S)<.B, C>. <-> (ARB /\ ASC)))
60	4, 9, 59	pm5.21nii 342	1 |- (A(R (x) S)<.B, C>. <-> (ARB /\ ASC))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  _Vcvv 2860   i^i cin 3209  <.cop 4562   class class class wbr 4640  1stc1st 4718   o. ccom 4722  `'ccnv 4772  2ndc2nd 4784   (x) 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