Theorem brtxp 5784

Description: Binary relationship over a tail cross product. (Contributed by SF, 11-Feb-2015.)

Assertion

Ref	Expression
brtxp	|- (A(R (x) S)B <-> E.xE.y(B = <.x, y>. /\ ARx /\ ASy))

Distinct variable groups:   x,A,y   x,B,y   x,R,y   x,S,y

Proof of Theorem brtxp

Dummy variables w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brin 4694	. . . 4 |- (A((`'1st o. R) i^i (`'2nd o. S))B <-> (A(`'1st o. R)B /\ A(`'2nd o. S)B))
2		brco 4884	. . . . 5 |- (A(`'1st o. R)B <-> E.x(ARx /\ x`'1stB))
3		brco 4884	. . . . 5 |- (A(`'2nd o. S)B <-> E.y(ASy /\ y`'2ndB))
4	2, 3	anbi12i 678	. . . 4 |- ((A(`'1st o. R)B /\ A(`'2nd o. S)B) <-> (E.x(ARx /\ x`'1stB) /\ E.y(ASy /\ y`'2ndB)))
5	1, 4	bitri 240	. . 3 |- (A((`'1st o. R) i^i (`'2nd o. S))B <-> (E.x(ARx /\ x`'1stB) /\ E.y(ASy /\ y`'2ndB)))
6		df-txp 5737	. . . 4 |- (R (x) S) = ((`'1st o. R) i^i (`'2nd o. S))
7	6	breqi 4646	. . 3 |- (A(R (x) S)B <-> A((`'1st o. R) i^i (`'2nd o. S))B)
8		eeanv 1913	. . 3 |- (E.xE.y((ARx /\ x`'1stB) /\ (ASy /\ y`'2ndB)) <-> (E.x(ARx /\ x`'1stB) /\ E.y(ASy /\ y`'2ndB)))
9	5, 7, 8	3bitr4i 268	. 2 |- (A(R (x) S)B <-> E.xE.y((ARx /\ x`'1stB) /\ (ASy /\ y`'2ndB)))
10		an4 797	. . . 4 |- (((ARx /\ x`'1stB) /\ (ASy /\ y`'2ndB)) <-> ((ARx /\ ASy) /\ (x`'1stB /\ y`'2ndB)))
11		ancom 437	. . . . 5 |- (((ARx /\ ASy) /\ B = <.x, y>.) <-> (B = <.x, y>. /\ (ARx /\ ASy)))
12		brcnv 4893	. . . . . . . . 9 |- (x`'1stB <-> B1stx)
13		vex 2863	. . . . . . . . . 10 |- x e. _V
14	13	br1st 4859	. . . . . . . . 9 |- (B1stx <-> E.z B = <.x, z>.)
15	12, 14	bitri 240	. . . . . . . 8 |- (x`'1stB <-> E.z B = <.x, z>.)
16		brcnv 4893	. . . . . . . . 9 |- (y`'2ndB <-> B2ndy)
17		vex 2863	. . . . . . . . . 10 |- y e. _V
18	17	br2nd 4860	. . . . . . . . 9 |- (B2ndy <-> E.w B = <.w, y>.)
19	16, 18	bitri 240	. . . . . . . 8 |- (y`'2ndB <-> E.w B = <.w, y>.)
20	15, 19	anbi12i 678	. . . . . . 7 |- ((x`'1stB /\ y`'2ndB) <-> (E.z B = <.x, z>. /\ E.w B = <.w, y>.))
21		eeanv 1913	. . . . . . 7 |- (E.zE.w(B = <.x, z>. /\ B = <.w, y>.) <-> (E.z B = <.x, z>. /\ E.w B = <.w, y>.))
22		eqtr2 2371	. . . . . . . . . . 11 |- ((B = <.x, z>. /\ B = <.w, y>.) -> <.x, z>. = <.w, y>.)
23		opth 4603	. . . . . . . . . . . . . 14 |- (<.x, z>. = <.w, y>. <-> (x = w /\ z = y))
24	23	simplbi 446	. . . . . . . . . . . . 13 |- (<.x, z>. = <.w, y>. -> x = w)
25	24	eqcomd 2358	. . . . . . . . . . . 12 |- (<.x, z>. = <.w, y>. -> w = x)
26	25	opeq1d 4585	. . . . . . . . . . 11 |- (<.x, z>. = <.w, y>. -> <.w, y>. = <.x, y>.)
27	22, 26	syl 15	. . . . . . . . . 10 |- ((B = <.x, z>. /\ B = <.w, y>.) -> <.w, y>. = <.x, y>.)
28		eqeq1 2359	. . . . . . . . . . 11 |- (B = <.w, y>. -> (B = <.x, y>. <-> <.w, y>. = <.x, y>.))
29	28	adantl 452	. . . . . . . . . 10 |- ((B = <.x, z>. /\ B = <.w, y>.) -> (B = <.x, y>. <-> <.w, y>. = <.x, y>.))
30	27, 29	mpbird 223	. . . . . . . . 9 |- ((B = <.x, z>. /\ B = <.w, y>.) -> B = <.x, y>.)
31	30	exlimivv 1635	. . . . . . . 8 |- (E.zE.w(B = <.x, z>. /\ B = <.w, y>.) -> B = <.x, y>.)
32		opeq2 4580	. . . . . . . . . . . 12 |- (z = y -> <.x, z>. = <.x, y>.)
33	32	eqeq2d 2364	. . . . . . . . . . 11 |- (z = y -> (B = <.x, z>. <-> B = <.x, y>.))
34		opeq1 4579	. . . . . . . . . . . 12 |- (w = x -> <.w, y>. = <.x, y>.)
35	34	eqeq2d 2364	. . . . . . . . . . 11 |- (w = x -> (B = <.w, y>. <-> B = <.x, y>.))
36	33, 35	bi2anan9 843	. . . . . . . . . 10 |- ((z = y /\ w = x) -> ((B = <.x, z>. /\ B = <.w, y>.) <-> (B = <.x, y>. /\ B = <.x, y>.)))
37	17, 13, 36	spc2ev 2948	. . . . . . . . 9 |- ((B = <.x, y>. /\ B = <.x, y>.) -> E.zE.w(B = <.x, z>. /\ B = <.w, y>.))
38	37	anidms 626	. . . . . . . 8 |- (B = <.x, y>. -> E.zE.w(B = <.x, z>. /\ B = <.w, y>.))
39	31, 38	impbii 180	. . . . . . 7 |- (E.zE.w(B = <.x, z>. /\ B = <.w, y>.) <-> B = <.x, y>.)
40	20, 21, 39	3bitr2i 264	. . . . . 6 |- ((x`'1stB /\ y`'2ndB) <-> B = <.x, y>.)
41	40	anbi2i 675	. . . . 5 |- (((ARx /\ ASy) /\ (x`'1stB /\ y`'2ndB)) <-> ((ARx /\ ASy) /\ B = <.x, y>.))
42		3anass 938	. . . . 5 |- ((B = <.x, y>. /\ ARx /\ ASy) <-> (B = <.x, y>. /\ (ARx /\ ASy)))
43	11, 41, 42	3bitr4i 268	. . . 4 |- (((ARx /\ ASy) /\ (x`'1stB /\ y`'2ndB)) <-> (B = <.x, y>. /\ ARx /\ ASy))
44	10, 43	bitri 240	. . 3 |- (((ARx /\ x`'1stB) /\ (ASy /\ y`'2ndB)) <-> (B = <.x, y>. /\ ARx /\ ASy))
45	44	2exbii 1583	. 2 |- (E.xE.y((ARx /\ x`'1stB) /\ (ASy /\ y`'2ndB)) <-> E.xE.y(B = <.x, y>. /\ ARx /\ ASy))
46	9, 45	bitri 240	1 |- (A(R (x) S)B <-> E.xE.y(B = <.x, y>. /\ ARx /\ ASy))

Syntax hints:   <-> wb 176   /\ wa 358   /\ w3a 934  E.wex 1541   = wceq 1642   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:  restxp  5787  oqelins4  5795  dmtxp  5803  fntxp  5805  brpprod  5840