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Theorem br1cosscnvxrn 37282
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 37195 . . . . . . 7 (𝐴𝑉 → [𝐴](𝑅𝑆) = {⟨𝑥, 𝑦⟩ ∣ (𝐴𝑅𝑥𝐴𝑆𝑦)})
2 ecxrn 37195 . . . . . . 7 (𝐵𝑊 → [𝐵](𝑅𝑆) = {⟨𝑥, 𝑦⟩ ∣ (𝐵𝑅𝑥𝐵𝑆𝑦)})
31, 2ineqan12d 4213 . . . . . 6 ((𝐴𝑉𝐵𝑊) → ([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) = ({⟨𝑥, 𝑦⟩ ∣ (𝐴𝑅𝑥𝐴𝑆𝑦)} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝐵𝑅𝑥𝐵𝑆𝑦)}))
4 inopab 5827 . . . . . 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 5214 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥𝐴𝑆𝑦) ∧ (𝐵𝑅𝑥𝐵𝑆𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦𝐵𝑆𝑦))}
85, 7eqtrdi 2789 . . . 4 ((𝐴𝑉𝐵𝑊) → ([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) = {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦𝐵𝑆𝑦))})
98neeq1d 3001 . . 3 ((𝐴𝑉𝐵𝑊) → (([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) ≠ ∅ ↔ {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦𝐵𝑆𝑦))} ≠ ∅))
10 opabn0 5552 . . . 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 37281 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑆)𝐵 ↔ ([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) ≠ ∅))
15 brcosscnv 37280 . . 3 ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥)))
16 brcosscnv 37280 . . 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 2941  cin 3946  c0 4321   class class class wbr 5147  {copab 5209  ccnv 5674  [cec 8697  cxrn 36980  ccoss 36981
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 5298  ax-nul 5305  ax-pr 5426  ax-un 7720
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 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fo 6546  df-fv 6548  df-1st 7970  df-2nd 7971  df-ec 8701  df-xrn 37179  df-coss 37219
This theorem is referenced by:  1cosscnvxrn  37283
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