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 35506
Description: Characterize a ternary relation over a range Cartesian product. Together with xrnss3v 35504, 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 35503 . . . 4 (𝑅𝑆) = (((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ 𝑆))
21breqi 5063 . . 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 5109 . . 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 5347 . . . . . 6 𝐵, 𝐶⟩ ∈ V
7 brcog 5730 . . . . . 6 ((𝐴𝑉 ∧ ⟨𝐵, 𝐶⟩ ∈ V) → (𝐴((1st ↾ (V × V)) ∘ 𝑅)⟨𝐵, 𝐶⟩ ↔ ∃𝑥(𝐴𝑅𝑥𝑥(1st ↾ (V × V))⟨𝐵, 𝐶⟩)))
86, 7mpan2 687 . . . . 5 (𝐴𝑉 → (𝐴((1st ↾ (V × V)) ∘ 𝑅)⟨𝐵, 𝐶⟩ ↔ ∃𝑥(𝐴𝑅𝑥𝑥(1st ↾ (V × V))⟨𝐵, 𝐶⟩)))
983ad2ant1 1125 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴((1st ↾ (V × V)) ∘ 𝑅)⟨𝐵, 𝐶⟩ ↔ ∃𝑥(𝐴𝑅𝑥𝑥(1st ↾ (V × V))⟨𝐵, 𝐶⟩)))
10 brcnvg 5743 . . . . . . . . 9 ((𝑥 ∈ V ∧ ⟨𝐵, 𝐶⟩ ∈ V) → (𝑥(1st ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥))
116, 10mpan2 687 . . . . . . . 8 (𝑥 ∈ V → (𝑥(1st ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥))
1211elv 3497 . . . . . . 7 (𝑥(1st ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥)
13 opelvvg 5588 . . . . . . . . . . 11 ((𝐵𝑊𝐶𝑋) → ⟨𝐵, 𝐶⟩ ∈ (V × V))
1413biantrurd 533 . . . . . . . . . 10 ((𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐶⟩1st 𝑥 ↔ (⟨𝐵, 𝐶⟩ ∈ (V × V) ∧ ⟨𝐵, 𝐶⟩1st 𝑥)))
15 brres 5853 . . . . . . . . . . 11 (𝑥 ∈ V → (⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥 ↔ (⟨𝐵, 𝐶⟩ ∈ (V × V) ∧ ⟨𝐵, 𝐶⟩1st 𝑥)))
1615elv 3497 . . . . . . . . . 10 (⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥 ↔ (⟨𝐵, 𝐶⟩ ∈ (V × V) ∧ ⟨𝐵, 𝐶⟩1st 𝑥))
1714, 16syl6rbbr 291 . . . . . . . . 9 ((𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥 ↔ ⟨𝐵, 𝐶⟩1st 𝑥))
18 br1steqg 7700 . . . . . . . . 9 ((𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐶⟩1st 𝑥𝑥 = 𝐵))
1917, 18bitrd 280 . . . . . . . 8 ((𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥𝑥 = 𝐵))
20193adant1 1122 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥𝑥 = 𝐵))
2112, 20syl5bb 284 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝑥(1st ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ 𝑥 = 𝐵))
2221anbi1cd 633 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴𝑅𝑥𝑥(1st ↾ (V × V))⟨𝐵, 𝐶⟩) ↔ (𝑥 = 𝐵𝐴𝑅𝑥)))
2322exbidv 1913 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∃𝑥(𝐴𝑅𝑥𝑥(1st ↾ (V × V))⟨𝐵, 𝐶⟩) ↔ ∃𝑥(𝑥 = 𝐵𝐴𝑅𝑥)))
24 breq2 5061 . . . . . 6 (𝑥 = 𝐵 → (𝐴𝑅𝑥𝐴𝑅𝐵))
2524ceqsexgv 3644 . . . . 5 (𝐵𝑊 → (∃𝑥(𝑥 = 𝐵𝐴𝑅𝑥) ↔ 𝐴𝑅𝐵))
26253ad2ant2 1126 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∃𝑥(𝑥 = 𝐵𝐴𝑅𝑥) ↔ 𝐴𝑅𝐵))
279, 23, 263bitrd 306 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴((1st ↾ (V × V)) ∘ 𝑅)⟨𝐵, 𝐶⟩ ↔ 𝐴𝑅𝐵))
28 brcog 5730 . . . . . 6 ((𝐴𝑉 ∧ ⟨𝐵, 𝐶⟩ ∈ V) → (𝐴((2nd ↾ (V × V)) ∘ 𝑆)⟨𝐵, 𝐶⟩ ↔ ∃𝑦(𝐴𝑆𝑦𝑦(2nd ↾ (V × V))⟨𝐵, 𝐶⟩)))
296, 28mpan2 687 . . . . 5 (𝐴𝑉 → (𝐴((2nd ↾ (V × V)) ∘ 𝑆)⟨𝐵, 𝐶⟩ ↔ ∃𝑦(𝐴𝑆𝑦𝑦(2nd ↾ (V × V))⟨𝐵, 𝐶⟩)))
30293ad2ant1 1125 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴((2nd ↾ (V × V)) ∘ 𝑆)⟨𝐵, 𝐶⟩ ↔ ∃𝑦(𝐴𝑆𝑦𝑦(2nd ↾ (V × V))⟨𝐵, 𝐶⟩)))
31 brcnvg 5743 . . . . . . . . 9 ((𝑦 ∈ V ∧ ⟨𝐵, 𝐶⟩ ∈ V) → (𝑦(2nd ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦))
326, 31mpan2 687 . . . . . . . 8 (𝑦 ∈ V → (𝑦(2nd ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦))
3332elv 3497 . . . . . . 7 (𝑦(2nd ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦)
3413biantrurd 533 . . . . . . . . . 10 ((𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐶⟩2nd 𝑦 ↔ (⟨𝐵, 𝐶⟩ ∈ (V × V) ∧ ⟨𝐵, 𝐶⟩2nd 𝑦)))
35 brres 5853 . . . . . . . . . . 11 (𝑦 ∈ V → (⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦 ↔ (⟨𝐵, 𝐶⟩ ∈ (V × V) ∧ ⟨𝐵, 𝐶⟩2nd 𝑦)))
3635elv 3497 . . . . . . . . . 10 (⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦 ↔ (⟨𝐵, 𝐶⟩ ∈ (V × V) ∧ ⟨𝐵, 𝐶⟩2nd 𝑦))
3734, 36syl6rbbr 291 . . . . . . . . 9 ((𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦 ↔ ⟨𝐵, 𝐶⟩2nd 𝑦))
38 br2ndeqg 7701 . . . . . . . . 9 ((𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐶⟩2nd 𝑦𝑦 = 𝐶))
3937, 38bitrd 280 . . . . . . . 8 ((𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦𝑦 = 𝐶))
40393adant1 1122 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦𝑦 = 𝐶))
4133, 40syl5bb 284 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝑦(2nd ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ 𝑦 = 𝐶))
4241anbi1cd 633 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴𝑆𝑦𝑦(2nd ↾ (V × V))⟨𝐵, 𝐶⟩) ↔ (𝑦 = 𝐶𝐴𝑆𝑦)))
4342exbidv 1913 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∃𝑦(𝐴𝑆𝑦𝑦(2nd ↾ (V × V))⟨𝐵, 𝐶⟩) ↔ ∃𝑦(𝑦 = 𝐶𝐴𝑆𝑦)))
44 breq2 5061 . . . . . 6 (𝑦 = 𝐶 → (𝐴𝑆𝑦𝐴𝑆𝐶))
4544ceqsexgv 3644 . . . . 5 (𝐶𝑋 → (∃𝑦(𝑦 = 𝐶𝐴𝑆𝑦) ↔ 𝐴𝑆𝐶))
46453ad2ant3 1127 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∃𝑦(𝑦 = 𝐶𝐴𝑆𝑦) ↔ 𝐴𝑆𝐶))
4730, 43, 463bitrd 306 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴((2nd ↾ (V × V)) ∘ 𝑆)⟨𝐵, 𝐶⟩ ↔ 𝐴𝑆𝐶))
4827, 47anbi12d 630 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴((1st ↾ (V × V)) ∘ 𝑅)⟨𝐵, 𝐶⟩ ∧ 𝐴((2nd ↾ (V × V)) ∘ 𝑆)⟨𝐵, 𝐶⟩) ↔ (𝐴𝑅𝐵𝐴𝑆𝐶)))
493, 5, 483bitrd 306 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴(𝑅𝑆)⟨𝐵, 𝐶⟩ ↔ (𝐴𝑅𝐵𝐴𝑆𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wex 1771  wcel 2105  Vcvv 3492  cin 3932  cop 4563   class class class wbr 5057   × cxp 5546  ccnv 5547  cres 5550  ccom 5552  1st c1st 7676  2nd c2nd 7677  cxrn 35333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fo 6354  df-fv 6356  df-1st 7678  df-2nd 7679  df-xrn 35503
This theorem is referenced by:  brxrn2  35507  dfxrn2  35508  brin2  35537  br1cossxrnres  35568
  Copyright terms: Public domain W3C validator