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Theorem brxrn 37244
Description: Characterize a ternary relation over a range Cartesian product. Together with xrnss3v 37242, 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 37241 . . . 4 (𝑅𝑆) = (((1st ↾ (V × V)) ∘ 𝑅) ∩ ((2nd ↾ (V × V)) ∘ 𝑆))
21breqi 5155 . . 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 5201 . . 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 5465 . . . . . 6 𝐵, 𝐶⟩ ∈ V
7 brcog 5867 . . . . . 6 ((𝐴𝑉 ∧ ⟨𝐵, 𝐶⟩ ∈ V) → (𝐴((1st ↾ (V × V)) ∘ 𝑅)⟨𝐵, 𝐶⟩ ↔ ∃𝑥(𝐴𝑅𝑥𝑥(1st ↾ (V × V))⟨𝐵, 𝐶⟩)))
86, 7mpan2 690 . . . . 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 5880 . . . . . . . . 9 ((𝑥 ∈ V ∧ ⟨𝐵, 𝐶⟩ ∈ V) → (𝑥(1st ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥))
116, 10mpan2 690 . . . . . . . 8 (𝑥 ∈ V → (𝑥(1st ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥))
1211elv 3481 . . . . . . 7 (𝑥(1st ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥)
13 brres 5989 . . . . . . . . . . 11 (𝑥 ∈ V → (⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥 ↔ (⟨𝐵, 𝐶⟩ ∈ (V × V) ∧ ⟨𝐵, 𝐶⟩1st 𝑥)))
1413elv 3481 . . . . . . . . . 10 (⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥 ↔ (⟨𝐵, 𝐶⟩ ∈ (V × V) ∧ ⟨𝐵, 𝐶⟩1st 𝑥))
15 opelvvg 5718 . . . . . . . . . . 11 ((𝐵𝑊𝐶𝑋) → ⟨𝐵, 𝐶⟩ ∈ (V × V))
1615biantrurd 534 . . . . . . . . . 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 7997 . . . . . . . . 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 1925 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∃𝑥(𝐴𝑅𝑥𝑥(1st ↾ (V × V))⟨𝐵, 𝐶⟩) ↔ ∃𝑥(𝑥 = 𝐵𝐴𝑅𝑥)))
24 breq2 5153 . . . . . 6 (𝑥 = 𝐵 → (𝐴𝑅𝑥𝐴𝑅𝐵))
2524ceqsexgv 3643 . . . . 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 5867 . . . . . 6 ((𝐴𝑉 ∧ ⟨𝐵, 𝐶⟩ ∈ V) → (𝐴((2nd ↾ (V × V)) ∘ 𝑆)⟨𝐵, 𝐶⟩ ↔ ∃𝑦(𝐴𝑆𝑦𝑦(2nd ↾ (V × V))⟨𝐵, 𝐶⟩)))
296, 28mpan2 690 . . . . 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 5880 . . . . . . . . 9 ((𝑦 ∈ V ∧ ⟨𝐵, 𝐶⟩ ∈ V) → (𝑦(2nd ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦))
326, 31mpan2 690 . . . . . . . 8 (𝑦 ∈ V → (𝑦(2nd ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦))
3332elv 3481 . . . . . . 7 (𝑦(2nd ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦)
34 brres 5989 . . . . . . . . . . 11 (𝑦 ∈ V → (⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦 ↔ (⟨𝐵, 𝐶⟩ ∈ (V × V) ∧ ⟨𝐵, 𝐶⟩2nd 𝑦)))
3534elv 3481 . . . . . . . . . 10 (⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦 ↔ (⟨𝐵, 𝐶⟩ ∈ (V × V) ∧ ⟨𝐵, 𝐶⟩2nd 𝑦))
3615biantrurd 534 . . . . . . . . . 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 7998 . . . . . . . . 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 1925 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∃𝑦(𝐴𝑆𝑦𝑦(2nd ↾ (V × V))⟨𝐵, 𝐶⟩) ↔ ∃𝑦(𝑦 = 𝐶𝐴𝑆𝑦)))
44 breq2 5153 . . . . . 6 (𝑦 = 𝐶 → (𝐴𝑆𝑦𝐴𝑆𝐶))
4544ceqsexgv 3643 . . . . 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 205  wa 397  w3a 1088   = wceq 1542  wex 1782  wcel 2107  Vcvv 3475  cin 3948  cop 4635   class class class wbr 5149   × cxp 5675  ccnv 5676  cres 5679  ccom 5681  1st c1st 7973  2nd c2nd 7974  cxrn 37042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-1st 7975  df-2nd 7976  df-xrn 37241
This theorem is referenced by:  brxrn2  37245  dfxrn2  37246  brin2  37279  br1cossxrnres  37318
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