Theorem brtxp 5784

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

Assertion

Ref	Expression
brtxp	⊢ (A(R ⊗ S)B ↔ ∃x∃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 ∘ R) ∩ (◡2nd ∘ S))B ↔ (A(◡1st ∘ R)B ∧ A(◡2nd ∘ S)B))
2		brco 4884	. . . . 5 ⊢ (A(◡1st ∘ R)B ↔ ∃x(ARx ∧ x◡1st B))
3		brco 4884	. . . . 5 ⊢ (A(◡2nd ∘ S)B ↔ ∃y(ASy ∧ y◡2nd B))
4	2, 3	anbi12i 678	. . . 4 ⊢ ((A(◡1st ∘ R)B ∧ A(◡2nd ∘ S)B) ↔ (∃x(ARx ∧ x◡1st B) ∧ ∃y(ASy ∧ y◡2nd B)))
5	1, 4	bitri 240	. . 3 ⊢ (A((◡1st ∘ R) ∩ (◡2nd ∘ S))B ↔ (∃x(ARx ∧ x◡1st B) ∧ ∃y(ASy ∧ y◡2nd B)))
6		df-txp 5737	. . . 4 ⊢ (R ⊗ S) = ((◡1st ∘ R) ∩ (◡2nd ∘ S))
7	6	breqi 4646	. . 3 ⊢ (A(R ⊗ S)B ↔ A((◡1st ∘ R) ∩ (◡2nd ∘ S))B)
8		eeanv 1913	. . 3 ⊢ (∃x∃y((ARx ∧ x◡1st B) ∧ (ASy ∧ y◡2nd B)) ↔ (∃x(ARx ∧ x◡1st B) ∧ ∃y(ASy ∧ y◡2nd B)))
9	5, 7, 8	3bitr4i 268	. 2 ⊢ (A(R ⊗ S)B ↔ ∃x∃y((ARx ∧ x◡1st B) ∧ (ASy ∧ y◡2nd B)))
10		an4 797	. . . 4 ⊢ (((ARx ∧ x◡1st B) ∧ (ASy ∧ y◡2nd B)) ↔ ((ARx ∧ ASy) ∧ (x◡1st B ∧ y◡2nd B)))
11		ancom 437	. . . . 5 ⊢ (((ARx ∧ ASy) ∧ B = ⟨x, y⟩) ↔ (B = ⟨x, y⟩ ∧ (ARx ∧ ASy)))
12		brcnv 4893	. . . . . . . . 9 ⊢ (x◡1st B ↔ B1st x)
13		vex 2863	. . . . . . . . . 10 ⊢ x ∈ V
14	13	br1st 4859	. . . . . . . . 9 ⊢ (B1st x ↔ ∃z B = ⟨x, z⟩)
15	12, 14	bitri 240	. . . . . . . 8 ⊢ (x◡1st B ↔ ∃z B = ⟨x, z⟩)
16		brcnv 4893	. . . . . . . . 9 ⊢ (y◡2nd B ↔ B2nd y)
17		vex 2863	. . . . . . . . . 10 ⊢ y ∈ V
18	17	br2nd 4860	. . . . . . . . 9 ⊢ (B2nd y ↔ ∃w B = ⟨w, y⟩)
19	16, 18	bitri 240	. . . . . . . 8 ⊢ (y◡2nd B ↔ ∃w B = ⟨w, y⟩)
20	15, 19	anbi12i 678	. . . . . . 7 ⊢ ((x◡1st B ∧ y◡2nd B) ↔ (∃z B = ⟨x, z⟩ ∧ ∃w B = ⟨w, y⟩))
21		eeanv 1913	. . . . . . 7 ⊢ (∃z∃w(B = ⟨x, z⟩ ∧ B = ⟨w, y⟩) ↔ (∃z B = ⟨x, z⟩ ∧ ∃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 ⊢ (∃z∃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⟩) → ∃z∃w(B = ⟨x, z⟩ ∧ B = ⟨w, y⟩))
38	37	anidms 626	. . . . . . . 8 ⊢ (B = ⟨x, y⟩ → ∃z∃w(B = ⟨x, z⟩ ∧ B = ⟨w, y⟩))
39	31, 38	impbii 180	. . . . . . 7 ⊢ (∃z∃w(B = ⟨x, z⟩ ∧ B = ⟨w, y⟩) ↔ B = ⟨x, y⟩)
40	20, 21, 39	3bitr2i 264	. . . . . 6 ⊢ ((x◡1st B ∧ y◡2nd B) ↔ B = ⟨x, y⟩)
41	40	anbi2i 675	. . . . 5 ⊢ (((ARx ∧ ASy) ∧ (x◡1st B ∧ y◡2nd B)) ↔ ((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◡1st B ∧ y◡2nd B)) ↔ (B = ⟨x, y⟩ ∧ ARx ∧ ASy))
44	10, 43	bitri 240	. . 3 ⊢ (((ARx ∧ x◡1st B) ∧ (ASy ∧ y◡2nd B)) ↔ (B = ⟨x, y⟩ ∧ ARx ∧ ASy))
45	44	2exbii 1583	. 2 ⊢ (∃x∃y((ARx ∧ x◡1st B) ∧ (ASy ∧ y◡2nd B)) ↔ ∃x∃y(B = ⟨x, y⟩ ∧ ARx ∧ ASy))
46	9, 45	bitri 240	1 ⊢ (A(R ⊗ S)B ↔ ∃x∃y(B = ⟨x, y⟩ ∧ ARx ∧ ASy))

This proof depends on syntax axioms:   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∩ cin 3209  ⟨cop 4562   class class class wbr 4640  1^st c1st 4718   ∘ ccom 4722  ^◡ccnv 4772  2^nd c2nd 4784   ⊗ ctxp 5736

This proof depends on 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 proof 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 used by:  restxp  5787  oqelins4  5795  dmtxp  5803  fntxp  5805  brpprod  5840