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Theorem brxrn 38375
Description: Characterize a ternary relation over a range Cartesian product. Together with xrnss3v 38373, 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 38372 . . . 4 (𝑅𝑆) = (((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ 𝑆))
21breqi 5149 . . 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 5195 . . 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 5469 . . . . . 6 𝐵, 𝐶⟩ ∈ V
7 brcog 5877 . . . . . 6 ((𝐴𝑉 ∧ ⟨𝐵, 𝐶⟩ ∈ V) → (𝐴((1st ↾ (V × V)) ∘ 𝑅)⟨𝐵, 𝐶⟩ ↔ ∃𝑥(𝐴𝑅𝑥𝑥(1st ↾ (V × V))⟨𝐵, 𝐶⟩)))
86, 7mpan2 691 . . . . 5 (𝐴𝑉 → (𝐴((1st ↾ (V × V)) ∘ 𝑅)⟨𝐵, 𝐶⟩ ↔ ∃𝑥(𝐴𝑅𝑥𝑥(1st ↾ (V × V))⟨𝐵, 𝐶⟩)))
983ad2ant1 1134 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴((1st ↾ (V × V)) ∘ 𝑅)⟨𝐵, 𝐶⟩ ↔ ∃𝑥(𝐴𝑅𝑥𝑥(1st ↾ (V × V))⟨𝐵, 𝐶⟩)))
10 brcnvg 5890 . . . . . . . . 9 ((𝑥 ∈ V ∧ ⟨𝐵, 𝐶⟩ ∈ V) → (𝑥(1st ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥))
116, 10mpan2 691 . . . . . . . 8 (𝑥 ∈ V → (𝑥(1st ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥))
1211elv 3485 . . . . . . 7 (𝑥(1st ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥)
13 brres 6004 . . . . . . . . . . 11 (𝑥 ∈ V → (⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥 ↔ (⟨𝐵, 𝐶⟩ ∈ (V × V) ∧ ⟨𝐵, 𝐶⟩1st 𝑥)))
1413elv 3485 . . . . . . . . . 10 (⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥 ↔ (⟨𝐵, 𝐶⟩ ∈ (V × V) ∧ ⟨𝐵, 𝐶⟩1st 𝑥))
15 opelvvg 5726 . . . . . . . . . . 11 ((𝐵𝑊𝐶𝑋) → ⟨𝐵, 𝐶⟩ ∈ (V × V))
1615biantrurd 532 . . . . . . . . . 10 ((𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐶⟩1st 𝑥 ↔ (⟨𝐵, 𝐶⟩ ∈ (V × V) ∧ ⟨𝐵, 𝐶⟩1st 𝑥)))
1714, 16bitr4id 290 . . . . . . . . 9 ((𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥 ↔ ⟨𝐵, 𝐶⟩1st 𝑥))
18 br1steqg 8036 . . . . . . . . 9 ((𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐶⟩1st 𝑥𝑥 = 𝐵))
1917, 18bitrd 279 . . . . . . . 8 ((𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥𝑥 = 𝐵))
20193adant1 1131 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥𝑥 = 𝐵))
2112, 20bitrid 283 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝑥(1st ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ 𝑥 = 𝐵))
2221anbi1cd 635 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴𝑅𝑥𝑥(1st ↾ (V × V))⟨𝐵, 𝐶⟩) ↔ (𝑥 = 𝐵𝐴𝑅𝑥)))
2322exbidv 1921 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∃𝑥(𝐴𝑅𝑥𝑥(1st ↾ (V × V))⟨𝐵, 𝐶⟩) ↔ ∃𝑥(𝑥 = 𝐵𝐴𝑅𝑥)))
24 breq2 5147 . . . . . 6 (𝑥 = 𝐵 → (𝐴𝑅𝑥𝐴𝑅𝐵))
2524ceqsexgv 3654 . . . . 5 (𝐵𝑊 → (∃𝑥(𝑥 = 𝐵𝐴𝑅𝑥) ↔ 𝐴𝑅𝐵))
26253ad2ant2 1135 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∃𝑥(𝑥 = 𝐵𝐴𝑅𝑥) ↔ 𝐴𝑅𝐵))
279, 23, 263bitrd 305 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴((1st ↾ (V × V)) ∘ 𝑅)⟨𝐵, 𝐶⟩ ↔ 𝐴𝑅𝐵))
28 brcog 5877 . . . . . 6 ((𝐴𝑉 ∧ ⟨𝐵, 𝐶⟩ ∈ V) → (𝐴((2nd ↾ (V × V)) ∘ 𝑆)⟨𝐵, 𝐶⟩ ↔ ∃𝑦(𝐴𝑆𝑦𝑦(2nd ↾ (V × V))⟨𝐵, 𝐶⟩)))
296, 28mpan2 691 . . . . 5 (𝐴𝑉 → (𝐴((2nd ↾ (V × V)) ∘ 𝑆)⟨𝐵, 𝐶⟩ ↔ ∃𝑦(𝐴𝑆𝑦𝑦(2nd ↾ (V × V))⟨𝐵, 𝐶⟩)))
30293ad2ant1 1134 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴((2nd ↾ (V × V)) ∘ 𝑆)⟨𝐵, 𝐶⟩ ↔ ∃𝑦(𝐴𝑆𝑦𝑦(2nd ↾ (V × V))⟨𝐵, 𝐶⟩)))
31 brcnvg 5890 . . . . . . . . 9 ((𝑦 ∈ V ∧ ⟨𝐵, 𝐶⟩ ∈ V) → (𝑦(2nd ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦))
326, 31mpan2 691 . . . . . . . 8 (𝑦 ∈ V → (𝑦(2nd ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦))
3332elv 3485 . . . . . . 7 (𝑦(2nd ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦)
34 brres 6004 . . . . . . . . . . 11 (𝑦 ∈ V → (⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦 ↔ (⟨𝐵, 𝐶⟩ ∈ (V × V) ∧ ⟨𝐵, 𝐶⟩2nd 𝑦)))
3534elv 3485 . . . . . . . . . 10 (⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦 ↔ (⟨𝐵, 𝐶⟩ ∈ (V × V) ∧ ⟨𝐵, 𝐶⟩2nd 𝑦))
3615biantrurd 532 . . . . . . . . . 10 ((𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐶⟩2nd 𝑦 ↔ (⟨𝐵, 𝐶⟩ ∈ (V × V) ∧ ⟨𝐵, 𝐶⟩2nd 𝑦)))
3735, 36bitr4id 290 . . . . . . . . 9 ((𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦 ↔ ⟨𝐵, 𝐶⟩2nd 𝑦))
38 br2ndeqg 8037 . . . . . . . . 9 ((𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐶⟩2nd 𝑦𝑦 = 𝐶))
3937, 38bitrd 279 . . . . . . . 8 ((𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦𝑦 = 𝐶))
40393adant1 1131 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦𝑦 = 𝐶))
4133, 40bitrid 283 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝑦(2nd ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ 𝑦 = 𝐶))
4241anbi1cd 635 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴𝑆𝑦𝑦(2nd ↾ (V × V))⟨𝐵, 𝐶⟩) ↔ (𝑦 = 𝐶𝐴𝑆𝑦)))
4342exbidv 1921 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∃𝑦(𝐴𝑆𝑦𝑦(2nd ↾ (V × V))⟨𝐵, 𝐶⟩) ↔ ∃𝑦(𝑦 = 𝐶𝐴𝑆𝑦)))
44 breq2 5147 . . . . . 6 (𝑦 = 𝐶 → (𝐴𝑆𝑦𝐴𝑆𝐶))
4544ceqsexgv 3654 . . . . 5 (𝐶𝑋 → (∃𝑦(𝑦 = 𝐶𝐴𝑆𝑦) ↔ 𝐴𝑆𝐶))
46453ad2ant3 1136 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∃𝑦(𝑦 = 𝐶𝐴𝑆𝑦) ↔ 𝐴𝑆𝐶))
4730, 43, 463bitrd 305 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴((2nd ↾ (V × V)) ∘ 𝑆)⟨𝐵, 𝐶⟩ ↔ 𝐴𝑆𝐶))
4827, 47anbi12d 632 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴((1st ↾ (V × V)) ∘ 𝑅)⟨𝐵, 𝐶⟩ ∧ 𝐴((2nd ↾ (V × V)) ∘ 𝑆)⟨𝐵, 𝐶⟩) ↔ (𝐴𝑅𝐵𝐴𝑆𝐶)))
493, 5, 483bitrd 305 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴(𝑅𝑆)⟨𝐵, 𝐶⟩ ↔ (𝐴𝑅𝐵𝐴𝑆𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  Vcvv 3480  cin 3950  cop 4632   class class class wbr 5143   × cxp 5683  ccnv 5684  cres 5687  ccom 5689  1st c1st 8012  2nd c2nd 8013  cxrn 38181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-1st 8014  df-2nd 8015  df-xrn 38372
This theorem is referenced by:  brxrn2  38376  dfxrn2  38377  brin2  38410  br1cossxrnres  38449
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