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Theorem br1cosscnvxrn 35873
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 35798 . . . . . . 7 (𝐴𝑉 → [𝐴](𝑅𝑆) = {⟨𝑥, 𝑦⟩ ∣ (𝐴𝑅𝑥𝐴𝑆𝑦)})
2 ecxrn 35798 . . . . . . 7 (𝐵𝑊 → [𝐵](𝑅𝑆) = {⟨𝑥, 𝑦⟩ ∣ (𝐵𝑅𝑥𝐵𝑆𝑦)})
31, 2ineqan12d 4144 . . . . . 6 ((𝐴𝑉𝐵𝑊) → ([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) = ({⟨𝑥, 𝑦⟩ ∣ (𝐴𝑅𝑥𝐴𝑆𝑦)} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝐵𝑅𝑥𝐵𝑆𝑦)}))
4 inopab 5669 . . . . . 6 ({⟨𝑥, 𝑦⟩ ∣ (𝐴𝑅𝑥𝐴𝑆𝑦)} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝐵𝑅𝑥𝐵𝑆𝑦)}) = {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥𝐴𝑆𝑦) ∧ (𝐵𝑅𝑥𝐵𝑆𝑦))}
53, 4eqtrdi 2852 . . . . 5 ((𝐴𝑉𝐵𝑊) → ([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) = {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥𝐴𝑆𝑦) ∧ (𝐵𝑅𝑥𝐵𝑆𝑦))})
6 an4 655 . . . . . 6 (((𝐴𝑅𝑥𝐴𝑆𝑦) ∧ (𝐵𝑅𝑥𝐵𝑆𝑦)) ↔ ((𝐴𝑅𝑥𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦𝐵𝑆𝑦)))
76opabbii 5100 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥𝐴𝑆𝑦) ∧ (𝐵𝑅𝑥𝐵𝑆𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦𝐵𝑆𝑦))}
85, 7eqtrdi 2852 . . . 4 ((𝐴𝑉𝐵𝑊) → ([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) = {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦𝐵𝑆𝑦))})
98neeq1d 3049 . . 3 ((𝐴𝑉𝐵𝑊) → (([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) ≠ ∅ ↔ {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦𝐵𝑆𝑦))} ≠ ∅))
10 opabn0 5408 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦𝐵𝑆𝑦))} ≠ ∅ ↔ ∃𝑥𝑦((𝐴𝑅𝑥𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦𝐵𝑆𝑦)))
11 exdistrv 1956 . . . 4 (∃𝑥𝑦((𝐴𝑅𝑥𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦𝐵𝑆𝑦)) ↔ (∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥) ∧ ∃𝑦(𝐴𝑆𝑦𝐵𝑆𝑦)))
1210, 11bitri 278 . . 3 ({⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦𝐵𝑆𝑦))} ≠ ∅ ↔ (∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥) ∧ ∃𝑦(𝐴𝑆𝑦𝐵𝑆𝑦)))
139, 12syl6bb 290 . 2 ((𝐴𝑉𝐵𝑊) → (([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) ≠ ∅ ↔ (∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥) ∧ ∃𝑦(𝐴𝑆𝑦𝐵𝑆𝑦))))
14 brcosscnv2 35872 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑆)𝐵 ↔ ([𝐴](𝑅𝑆) ∩ [𝐵](𝑅𝑆)) ≠ ∅))
15 brcosscnv 35871 . . 3 ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥)))
16 brcosscnv 35871 . . 3 ((𝐴𝑉𝐵𝑊) → (𝐴𝑆𝐵 ↔ ∃𝑦(𝐴𝑆𝑦𝐵𝑆𝑦)))
1715, 16anbi12d 633 . 2 ((𝐴𝑉𝐵𝑊) → ((𝐴𝑅𝐵𝐴𝑆𝐵) ↔ (∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥) ∧ ∃𝑦(𝐴𝑆𝑦𝐵𝑆𝑦))))
1813, 14, 173bitr4d 314 1 ((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑆)𝐵 ↔ (𝐴𝑅𝐵𝐴𝑆𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wex 1781  wcel 2112  wne 2990  cin 3883  c0 4246   class class class wbr 5033  {copab 5095  ccnv 5522  [cec 8274  cxrn 35611  ccoss 35612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fo 6334  df-fv 6336  df-1st 7675  df-2nd 7676  df-ec 8278  df-xrn 35782  df-coss 35818
This theorem is referenced by:  1cosscnvxrn  35874
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