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Theorem br1cosscnvxrn 38476
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 38389 . . . . . . 7 (𝐴𝑉 → [𝐴](𝑅𝑆) = {⟨𝑥, 𝑦⟩ ∣ (𝐴𝑅𝑥𝐴𝑆𝑦)})
2 ecxrn 38389 . . . . . . 7 (𝐵𝑊 → [𝐵](𝑅𝑆) = {⟨𝑥, 𝑦⟩ ∣ (𝐵𝑅𝑥𝐵𝑆𝑦)})
31, 2ineqan12d 4221 . . . . . 6 ((𝐴𝑉𝐵𝑊) → ([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) = ({⟨𝑥, 𝑦⟩ ∣ (𝐴𝑅𝑥𝐴𝑆𝑦)} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝐵𝑅𝑥𝐵𝑆𝑦)}))
4 inopab 5838 . . . . . 6 ({⟨𝑥, 𝑦⟩ ∣ (𝐴𝑅𝑥𝐴𝑆𝑦)} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝐵𝑅𝑥𝐵𝑆𝑦)}) = {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥𝐴𝑆𝑦) ∧ (𝐵𝑅𝑥𝐵𝑆𝑦))}
53, 4eqtrdi 2792 . . . . 5 ((𝐴𝑉𝐵𝑊) → ([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) = {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥𝐴𝑆𝑦) ∧ (𝐵𝑅𝑥𝐵𝑆𝑦))})
6 an4 656 . . . . . 6 (((𝐴𝑅𝑥𝐴𝑆𝑦) ∧ (𝐵𝑅𝑥𝐵𝑆𝑦)) ↔ ((𝐴𝑅𝑥𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦𝐵𝑆𝑦)))
76opabbii 5209 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥𝐴𝑆𝑦) ∧ (𝐵𝑅𝑥𝐵𝑆𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦𝐵𝑆𝑦))}
85, 7eqtrdi 2792 . . . 4 ((𝐴𝑉𝐵𝑊) → ([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) = {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦𝐵𝑆𝑦))})
98neeq1d 2999 . . 3 ((𝐴𝑉𝐵𝑊) → (([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) ≠ ∅ ↔ {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦𝐵𝑆𝑦))} ≠ ∅))
10 opabn0 5557 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦𝐵𝑆𝑦))} ≠ ∅ ↔ ∃𝑥𝑦((𝐴𝑅𝑥𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦𝐵𝑆𝑦)))
11 exdistrv 1954 . . . 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 38475 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑆)𝐵 ↔ ([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) ≠ ∅))
15 brcosscnv 38474 . . 3 ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥)))
16 brcosscnv 38474 . . 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 206  wa 395  wex 1778  wcel 2107  wne 2939  cin 3949  c0 4332   class class class wbr 5142  {copab 5204  ccnv 5683  [cec 8744  cxrn 38182  ccoss 38183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fo 6566  df-fv 6568  df-1st 8015  df-2nd 8016  df-ec 8748  df-xrn 38373  df-coss 38413
This theorem is referenced by:  1cosscnvxrn  38477
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