Theorem brxrn 39060

Description: Characterize a ternary relation over a range Cartesian product. Together with xrnss3v 39058, this characterizes elementhood in a range cross. (Contributed by Peter Mazsa, 27-Jun-2021.)

Assertion

Ref	Expression
brxrn	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴(𝑅 ⋉ 𝑆)⟨𝐵, 𝐶⟩ ↔ (𝐴𝑅𝐵 ∧ 𝐴𝑆𝐶)))

Proof of Theorem brxrn

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

Step	Hyp	Ref	Expression
1		df-xrn 39057	. . . 4 ⊢ (𝑅 ⋉ 𝑆) = ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆))
2	1	breqi 5114	. . 3 ⊢ (𝐴(𝑅 ⋉ 𝑆)⟨𝐵, 𝐶⟩ ↔ 𝐴((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆))⟨𝐵, 𝐶⟩)
3	2	a1i 11	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴(𝑅 ⋉ 𝑆)⟨𝐵, 𝐶⟩ ↔ 𝐴((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆))⟨𝐵, 𝐶⟩))
4		brin 5162	. . 3 ⊢ (𝐴((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆))⟨𝐵, 𝐶⟩ ↔ (𝐴(◡(1st ↾ (V × V)) ∘ 𝑅)⟨𝐵, 𝐶⟩ ∧ 𝐴(◡(2nd ↾ (V × V)) ∘ 𝑆)⟨𝐵, 𝐶⟩))
5	4	a1i 11	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆))⟨𝐵, 𝐶⟩ ↔ (𝐴(◡(1st ↾ (V × V)) ∘ 𝑅)⟨𝐵, 𝐶⟩ ∧ 𝐴(◡(2nd ↾ (V × V)) ∘ 𝑆)⟨𝐵, 𝐶⟩)))
6		opex 5444	. . . . . 6 ⊢ ⟨𝐵, 𝐶⟩ ∈ V
7		brcog 5851	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ⟨𝐵, 𝐶⟩ ∈ V) → (𝐴(◡(1st ↾ (V × V)) ∘ 𝑅)⟨𝐵, 𝐶⟩ ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝑥◡(1st ↾ (V × V))⟨𝐵, 𝐶⟩)))
8	6, 7	mpan2 703	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐴(◡(1st ↾ (V × V)) ∘ 𝑅)⟨𝐵, 𝐶⟩ ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝑥◡(1st ↾ (V × V))⟨𝐵, 𝐶⟩)))
9	8	3ad2ant1 1150	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴(◡(1st ↾ (V × V)) ∘ 𝑅)⟨𝐵, 𝐶⟩ ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝑥◡(1st ↾ (V × V))⟨𝐵, 𝐶⟩)))
10		brcnvg 5864	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ ⟨𝐵, 𝐶⟩ ∈ V) → (𝑥◡(1st ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥))
11	6, 10	mpan2 703	. . . . . . . 8 ⊢ (𝑥 ∈ V → (𝑥◡(1st ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥))
12	11	elv 3459	. . . . . . 7 ⊢ (𝑥◡(1st ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥)
13		brres 5984	. . . . . . . . . . 11 ⊢ (𝑥 ∈ V → (⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥 ↔ (⟨𝐵, 𝐶⟩ ∈ (V × V) ∧ ⟨𝐵, 𝐶⟩1st 𝑥)))
14	13	elv 3459	. . . . . . . . . 10 ⊢ (⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥 ↔ (⟨𝐵, 𝐶⟩ ∈ (V × V) ∧ ⟨𝐵, 𝐶⟩1st 𝑥))
15		opelvvg 5701	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ⟨𝐵, 𝐶⟩ ∈ (V × V))
16	15	biantrurd 541	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐵, 𝐶⟩1st 𝑥 ↔ (⟨𝐵, 𝐶⟩ ∈ (V × V) ∧ ⟨𝐵, 𝐶⟩1st 𝑥)))
17	14, 16	bitr4id 293	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥 ↔ ⟨𝐵, 𝐶⟩1st 𝑥))
18		br1steqg 8006	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐵, 𝐶⟩1st 𝑥 ↔ 𝑥 = 𝐵))
19	17, 18	bitrd 282	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥 ↔ 𝑥 = 𝐵))
20	19	3adant1 1147	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥 ↔ 𝑥 = 𝐵))
21	12, 20	bitrid 286	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝑥◡(1st ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ 𝑥 = 𝐵))
22	21	anbi1cd 646	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴𝑅𝑥 ∧ 𝑥◡(1st ↾ (V × V))⟨𝐵, 𝐶⟩) ↔ (𝑥 = 𝐵 ∧ 𝐴𝑅𝑥)))
23	22	exbidv 1950	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∃𝑥(𝐴𝑅𝑥 ∧ 𝑥◡(1st ↾ (V × V))⟨𝐵, 𝐶⟩) ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝐴𝑅𝑥)))
24		breq2 5112	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴𝑅𝑥 ↔ 𝐴𝑅𝐵))
25	24	ceqsexgv 3612	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → (∃𝑥(𝑥 = 𝐵 ∧ 𝐴𝑅𝑥) ↔ 𝐴𝑅𝐵))
26	25	3ad2ant2 1151	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∃𝑥(𝑥 = 𝐵 ∧ 𝐴𝑅𝑥) ↔ 𝐴𝑅𝐵))
27	9, 23, 26	3bitrd 308	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴(◡(1st ↾ (V × V)) ∘ 𝑅)⟨𝐵, 𝐶⟩ ↔ 𝐴𝑅𝐵))
28		brcog 5851	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ⟨𝐵, 𝐶⟩ ∈ V) → (𝐴(◡(2nd ↾ (V × V)) ∘ 𝑆)⟨𝐵, 𝐶⟩ ↔ ∃𝑦(𝐴𝑆𝑦 ∧ 𝑦◡(2nd ↾ (V × V))⟨𝐵, 𝐶⟩)))
29	6, 28	mpan2 703	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐴(◡(2nd ↾ (V × V)) ∘ 𝑆)⟨𝐵, 𝐶⟩ ↔ ∃𝑦(𝐴𝑆𝑦 ∧ 𝑦◡(2nd ↾ (V × V))⟨𝐵, 𝐶⟩)))
30	29	3ad2ant1 1150	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴(◡(2nd ↾ (V × V)) ∘ 𝑆)⟨𝐵, 𝐶⟩ ↔ ∃𝑦(𝐴𝑆𝑦 ∧ 𝑦◡(2nd ↾ (V × V))⟨𝐵, 𝐶⟩)))
31		brcnvg 5864	. . . . . . . . 9 ⊢ ((𝑦 ∈ V ∧ ⟨𝐵, 𝐶⟩ ∈ V) → (𝑦◡(2nd ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦))
32	6, 31	mpan2 703	. . . . . . . 8 ⊢ (𝑦 ∈ V → (𝑦◡(2nd ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦))
33	32	elv 3459	. . . . . . 7 ⊢ (𝑦◡(2nd ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦)
34		brres 5984	. . . . . . . . . . 11 ⊢ (𝑦 ∈ V → (⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦 ↔ (⟨𝐵, 𝐶⟩ ∈ (V × V) ∧ ⟨𝐵, 𝐶⟩2nd 𝑦)))
35	34	elv 3459	. . . . . . . . . 10 ⊢ (⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦 ↔ (⟨𝐵, 𝐶⟩ ∈ (V × V) ∧ ⟨𝐵, 𝐶⟩2nd 𝑦))
36	15	biantrurd 541	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐵, 𝐶⟩2nd 𝑦 ↔ (⟨𝐵, 𝐶⟩ ∈ (V × V) ∧ ⟨𝐵, 𝐶⟩2nd 𝑦)))
37	35, 36	bitr4id 293	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦 ↔ ⟨𝐵, 𝐶⟩2nd 𝑦))
38		br2ndeqg 8007	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐵, 𝐶⟩2nd 𝑦 ↔ 𝑦 = 𝐶))
39	37, 38	bitrd 282	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦 ↔ 𝑦 = 𝐶))
40	39	3adant1 1147	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦 ↔ 𝑦 = 𝐶))
41	33, 40	bitrid 286	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝑦◡(2nd ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ 𝑦 = 𝐶))
42	41	anbi1cd 646	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴𝑆𝑦 ∧ 𝑦◡(2nd ↾ (V × V))⟨𝐵, 𝐶⟩) ↔ (𝑦 = 𝐶 ∧ 𝐴𝑆𝑦)))
43	42	exbidv 1950	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∃𝑦(𝐴𝑆𝑦 ∧ 𝑦◡(2nd ↾ (V × V))⟨𝐵, 𝐶⟩) ↔ ∃𝑦(𝑦 = 𝐶 ∧ 𝐴𝑆𝑦)))
44		breq2 5112	. . . . . 6 ⊢ (𝑦 = 𝐶 → (𝐴𝑆𝑦 ↔ 𝐴𝑆𝐶))
45	44	ceqsexgv 3612	. . . . 5 ⊢ (𝐶 ∈ 𝑋 → (∃𝑦(𝑦 = 𝐶 ∧ 𝐴𝑆𝑦) ↔ 𝐴𝑆𝐶))
46	45	3ad2ant3 1152	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∃𝑦(𝑦 = 𝐶 ∧ 𝐴𝑆𝑦) ↔ 𝐴𝑆𝐶))
47	30, 43, 46	3bitrd 308	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴(◡(2nd ↾ (V × V)) ∘ 𝑆)⟨𝐵, 𝐶⟩ ↔ 𝐴𝑆𝐶))
48	27, 47	anbi12d 643	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴(◡(1st ↾ (V × V)) ∘ 𝑅)⟨𝐵, 𝐶⟩ ∧ 𝐴(◡(2nd ↾ (V × V)) ∘ 𝑆)⟨𝐵, 𝐶⟩) ↔ (𝐴𝑅𝐵 ∧ 𝐴𝑆𝐶)))
49	3, 5, 48	3bitrd 308	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴(𝑅 ⋉ 𝑆)⟨𝐵, 𝐶⟩ ↔ (𝐴𝑅𝐵 ∧ 𝐴𝑆𝐶)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1102   = wceq 1569  ∃wex 1808   ∈ wcel 2142  Vcvv 3454   ∩ cin 3903  ⟨cop 4594   class class class wbr 5108   × cxp 5658  ^◡ccnv 5659   ↾ cres 5662   ∘ ccom 5664  1^st c1st 7982  2^nd c2nd 7983   ⋉ cxrn 38851

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544  df-1st 7984  df-2nd 7985  df-xrn 39057

This theorem is used by:  brxrn2  39061  dfxrn2  39062  brxrncnvep  39063  ecxrn2  39085  brin2  39115  br1cossxrnres  39215