Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brxrn Structured version   Visualization version   GIF version

Theorem brxrn 34451
Description: Characterize a ternary relation over a range Cartesian product. Together with xrnss3v 34449, this characterizes elementhood in a range cross. (Contributed by Peter Mazsa, 27-Jun-2021.)
Assertion
Ref Expression
brxrn ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴(𝑅𝑆)⟨𝐵, 𝐶⟩ ↔ (𝐴𝑅𝐵𝐴𝑆𝐶)))

Proof of Theorem brxrn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-xrn 34448 . . . 4 (𝑅𝑆) = (((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ 𝑆))
21breqi 4802 . . 3 (𝐴(𝑅𝑆)⟨𝐵, 𝐶⟩ ↔ 𝐴(((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ 𝑆))⟨𝐵, 𝐶⟩)
32a1i 11 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴(𝑅𝑆)⟨𝐵, 𝐶⟩ ↔ 𝐴(((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ 𝑆))⟨𝐵, 𝐶⟩))
4 brin 4848 . . 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⟩))
54a1i 11 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (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 5073 . . . . . 6 𝐵, 𝐶⟩ ∈ V
7 brcog 5436 . . . . . 6 ((𝐴𝑉 ∧ ⟨𝐵, 𝐶⟩ ∈ V) → (𝐴((1st ↾ (V × V)) ∘ 𝑅)⟨𝐵, 𝐶⟩ ↔ ∃𝑥(𝐴𝑅𝑥𝑥(1st ↾ (V × V))⟨𝐵, 𝐶⟩)))
86, 7mpan2 709 . . . . 5 (𝐴𝑉 → (𝐴((1st ↾ (V × V)) ∘ 𝑅)⟨𝐵, 𝐶⟩ ↔ ∃𝑥(𝐴𝑅𝑥𝑥(1st ↾ (V × V))⟨𝐵, 𝐶⟩)))
983ad2ant1 1127 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴((1st ↾ (V × V)) ∘ 𝑅)⟨𝐵, 𝐶⟩ ↔ ∃𝑥(𝐴𝑅𝑥𝑥(1st ↾ (V × V))⟨𝐵, 𝐶⟩)))
10 brcnvg 5450 . . . . . . . . 9 ((𝑥 ∈ V ∧ ⟨𝐵, 𝐶⟩ ∈ V) → (𝑥(1st ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥))
116, 10mpan2 709 . . . . . . . 8 (𝑥 ∈ V → (𝑥(1st ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥))
1211elv 34301 . . . . . . 7 (𝑥(1st ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥)
13 opelvvg 5314 . . . . . . . . . . 11 ((𝐵𝑊𝐶𝑋) → ⟨𝐵, 𝐶⟩ ∈ (V × V))
1413biantrurd 530 . . . . . . . . . 10 ((𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐶⟩1st 𝑥 ↔ (⟨𝐵, 𝐶⟩ ∈ (V × V) ∧ ⟨𝐵, 𝐶⟩1st 𝑥)))
15 brresALTV 34348 . . . . . . . . . . 11 (𝑥 ∈ V → (⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥 ↔ (⟨𝐵, 𝐶⟩ ∈ (V × V) ∧ ⟨𝐵, 𝐶⟩1st 𝑥)))
1615elv 34301 . . . . . . . . . 10 (⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥 ↔ (⟨𝐵, 𝐶⟩ ∈ (V × V) ∧ ⟨𝐵, 𝐶⟩1st 𝑥))
1714, 16syl6rbbr 279 . . . . . . . . 9 ((𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥 ↔ ⟨𝐵, 𝐶⟩1st 𝑥))
18 br1steqg 7347 . . . . . . . . 9 ((𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐶⟩1st 𝑥𝑥 = 𝐵))
1917, 18bitrd 268 . . . . . . . 8 ((𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥𝑥 = 𝐵))
20193adant1 1124 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥𝑥 = 𝐵))
2112, 20syl5bb 272 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝑥(1st ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ 𝑥 = 𝐵))
2221anbi1cd 34315 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴𝑅𝑥𝑥(1st ↾ (V × V))⟨𝐵, 𝐶⟩) ↔ (𝑥 = 𝐵𝐴𝑅𝑥)))
2322exbidv 1991 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∃𝑥(𝐴𝑅𝑥𝑥(1st ↾ (V × V))⟨𝐵, 𝐶⟩) ↔ ∃𝑥(𝑥 = 𝐵𝐴𝑅𝑥)))
24 breq2 4800 . . . . . 6 (𝑥 = 𝐵 → (𝐴𝑅𝑥𝐴𝑅𝐵))
2524ceqsexgv 3466 . . . . 5 (𝐵𝑊 → (∃𝑥(𝑥 = 𝐵𝐴𝑅𝑥) ↔ 𝐴𝑅𝐵))
26253ad2ant2 1128 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∃𝑥(𝑥 = 𝐵𝐴𝑅𝑥) ↔ 𝐴𝑅𝐵))
279, 23, 263bitrd 294 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴((1st ↾ (V × V)) ∘ 𝑅)⟨𝐵, 𝐶⟩ ↔ 𝐴𝑅𝐵))
28 brcog 5436 . . . . . 6 ((𝐴𝑉 ∧ ⟨𝐵, 𝐶⟩ ∈ V) → (𝐴((2nd ↾ (V × V)) ∘ 𝑆)⟨𝐵, 𝐶⟩ ↔ ∃𝑦(𝐴𝑆𝑦𝑦(2nd ↾ (V × V))⟨𝐵, 𝐶⟩)))
296, 28mpan2 709 . . . . 5 (𝐴𝑉 → (𝐴((2nd ↾ (V × V)) ∘ 𝑆)⟨𝐵, 𝐶⟩ ↔ ∃𝑦(𝐴𝑆𝑦𝑦(2nd ↾ (V × V))⟨𝐵, 𝐶⟩)))
30293ad2ant1 1127 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴((2nd ↾ (V × V)) ∘ 𝑆)⟨𝐵, 𝐶⟩ ↔ ∃𝑦(𝐴𝑆𝑦𝑦(2nd ↾ (V × V))⟨𝐵, 𝐶⟩)))
31 brcnvg 5450 . . . . . . . . 9 ((𝑦 ∈ V ∧ ⟨𝐵, 𝐶⟩ ∈ V) → (𝑦(2nd ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦))
326, 31mpan2 709 . . . . . . . 8 (𝑦 ∈ V → (𝑦(2nd ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦))
3332elv 34301 . . . . . . 7 (𝑦(2nd ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦)
3413biantrurd 530 . . . . . . . . . 10 ((𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐶⟩2nd 𝑦 ↔ (⟨𝐵, 𝐶⟩ ∈ (V × V) ∧ ⟨𝐵, 𝐶⟩2nd 𝑦)))
35 brresALTV 34348 . . . . . . . . . . 11 (𝑦 ∈ V → (⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦 ↔ (⟨𝐵, 𝐶⟩ ∈ (V × V) ∧ ⟨𝐵, 𝐶⟩2nd 𝑦)))
3635elv 34301 . . . . . . . . . 10 (⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦 ↔ (⟨𝐵, 𝐶⟩ ∈ (V × V) ∧ ⟨𝐵, 𝐶⟩2nd 𝑦))
3734, 36syl6rbbr 279 . . . . . . . . 9 ((𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦 ↔ ⟨𝐵, 𝐶⟩2nd 𝑦))
38 br2ndeqg 7348 . . . . . . . . 9 ((𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐶⟩2nd 𝑦𝑦 = 𝐶))
3937, 38bitrd 268 . . . . . . . 8 ((𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦𝑦 = 𝐶))
40393adant1 1124 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦𝑦 = 𝐶))
4133, 40syl5bb 272 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝑦(2nd ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ 𝑦 = 𝐶))
4241anbi1cd 34315 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴𝑆𝑦𝑦(2nd ↾ (V × V))⟨𝐵, 𝐶⟩) ↔ (𝑦 = 𝐶𝐴𝑆𝑦)))
4342exbidv 1991 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∃𝑦(𝐴𝑆𝑦𝑦(2nd ↾ (V × V))⟨𝐵, 𝐶⟩) ↔ ∃𝑦(𝑦 = 𝐶𝐴𝑆𝑦)))
44 breq2 4800 . . . . . 6 (𝑦 = 𝐶 → (𝐴𝑆𝑦𝐴𝑆𝐶))
4544ceqsexgv 3466 . . . . 5 (𝐶𝑋 → (∃𝑦(𝑦 = 𝐶𝐴𝑆𝑦) ↔ 𝐴𝑆𝐶))
46453ad2ant3 1129 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∃𝑦(𝑦 = 𝐶𝐴𝑆𝑦) ↔ 𝐴𝑆𝐶))
4730, 43, 463bitrd 294 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴((2nd ↾ (V × V)) ∘ 𝑆)⟨𝐵, 𝐶⟩ ↔ 𝐴𝑆𝐶))
4827, 47anbi12d 749 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴((1st ↾ (V × V)) ∘ 𝑅)⟨𝐵, 𝐶⟩ ∧ 𝐴((2nd ↾ (V × V)) ∘ 𝑆)⟨𝐵, 𝐶⟩) ↔ (𝐴𝑅𝐵𝐴𝑆𝐶)))
493, 5, 483bitrd 294 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴(𝑅𝑆)⟨𝐵, 𝐶⟩ ↔ (𝐴𝑅𝐵𝐴𝑆𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1624  wex 1845  wcel 2131  Vcvv 3332  cin 3706  cop 4319   class class class wbr 4796   × cxp 5256  ccnv 5257  cres 5260  ccom 5262  1st c1st 7323  2nd c2nd 7324  cxrn 34287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-fo 6047  df-fv 6049  df-1st 7325  df-2nd 7326  df-xrn 34448
This theorem is referenced by:  brxrn2  34452  dfxrn2  34453  brin2  34482  br1cossxrnres  34513
  Copyright terms: Public domain W3C validator