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Theorem br1cosscnvxrn 36982
Description: 𝐴 and 𝐵 are cosets by the converse range Cartesian product: a binary relation. (Contributed by Peter Mazsa, 19-Apr-2020.) (Revised by Peter Mazsa, 21-Sep-2021.)
Assertion
Ref Expression
br1cosscnvxrn ((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑆)𝐵 ↔ (𝐴𝑅𝐵𝐴𝑆𝐵)))

Proof of Theorem br1cosscnvxrn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ecxrn 36895 . . . . . . 7 (𝐴𝑉 → [𝐴](𝑅𝑆) = {⟨𝑥, 𝑦⟩ ∣ (𝐴𝑅𝑥𝐴𝑆𝑦)})
2 ecxrn 36895 . . . . . . 7 (𝐵𝑊 → [𝐵](𝑅𝑆) = {⟨𝑥, 𝑦⟩ ∣ (𝐵𝑅𝑥𝐵𝑆𝑦)})
31, 2ineqan12d 4175 . . . . . 6 ((𝐴𝑉𝐵𝑊) → ([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) = ({⟨𝑥, 𝑦⟩ ∣ (𝐴𝑅𝑥𝐴𝑆𝑦)} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝐵𝑅𝑥𝐵𝑆𝑦)}))
4 inopab 5786 . . . . . 6 ({⟨𝑥, 𝑦⟩ ∣ (𝐴𝑅𝑥𝐴𝑆𝑦)} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝐵𝑅𝑥𝐵𝑆𝑦)}) = {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥𝐴𝑆𝑦) ∧ (𝐵𝑅𝑥𝐵𝑆𝑦))}
53, 4eqtrdi 2789 . . . . 5 ((𝐴𝑉𝐵𝑊) → ([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) = {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥𝐴𝑆𝑦) ∧ (𝐵𝑅𝑥𝐵𝑆𝑦))})
6 an4 655 . . . . . 6 (((𝐴𝑅𝑥𝐴𝑆𝑦) ∧ (𝐵𝑅𝑥𝐵𝑆𝑦)) ↔ ((𝐴𝑅𝑥𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦𝐵𝑆𝑦)))
76opabbii 5173 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥𝐴𝑆𝑦) ∧ (𝐵𝑅𝑥𝐵𝑆𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦𝐵𝑆𝑦))}
85, 7eqtrdi 2789 . . . 4 ((𝐴𝑉𝐵𝑊) → ([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) = {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦𝐵𝑆𝑦))})
98neeq1d 3000 . . 3 ((𝐴𝑉𝐵𝑊) → (([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) ≠ ∅ ↔ {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦𝐵𝑆𝑦))} ≠ ∅))
10 opabn0 5511 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦𝐵𝑆𝑦))} ≠ ∅ ↔ ∃𝑥𝑦((𝐴𝑅𝑥𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦𝐵𝑆𝑦)))
11 exdistrv 1960 . . . 4 (∃𝑥𝑦((𝐴𝑅𝑥𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦𝐵𝑆𝑦)) ↔ (∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥) ∧ ∃𝑦(𝐴𝑆𝑦𝐵𝑆𝑦)))
1210, 11bitri 275 . . 3 ({⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦𝐵𝑆𝑦))} ≠ ∅ ↔ (∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥) ∧ ∃𝑦(𝐴𝑆𝑦𝐵𝑆𝑦)))
139, 12bitrdi 287 . 2 ((𝐴𝑉𝐵𝑊) → (([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) ≠ ∅ ↔ (∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥) ∧ ∃𝑦(𝐴𝑆𝑦𝐵𝑆𝑦))))
14 brcosscnv2 36981 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑆)𝐵 ↔ ([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) ≠ ∅))
15 brcosscnv 36980 . . 3 ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥)))
16 brcosscnv 36980 . . 3 ((𝐴𝑉𝐵𝑊) → (𝐴𝑆𝐵 ↔ ∃𝑦(𝐴𝑆𝑦𝐵𝑆𝑦)))
1715, 16anbi12d 632 . 2 ((𝐴𝑉𝐵𝑊) → ((𝐴𝑅𝐵𝐴𝑆𝐵) ↔ (∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥) ∧ ∃𝑦(𝐴𝑆𝑦𝐵𝑆𝑦))))
1813, 14, 173bitr4d 311 1 ((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑆)𝐵 ↔ (𝐴𝑅𝐵𝐴𝑆𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wex 1782  wcel 2107  wne 2940  cin 3910  c0 4283   class class class wbr 5106  {copab 5168  ccnv 5633  [cec 8649  cxrn 36679  ccoss 36680
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 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
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 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fo 6503  df-fv 6505  df-1st 7922  df-2nd 7923  df-ec 8653  df-xrn 36879  df-coss 36919
This theorem is referenced by:  1cosscnvxrn  36983
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