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Theorem 1cosscnvxrn 37871
Description: Cosets by the converse range Cartesian product. (Contributed by Peter Mazsa, 19-Apr-2020.) (Revised by Peter Mazsa, 21-Sep-2021.)
Assertion
Ref Expression
1cosscnvxrn (𝐴𝐵) = ( ≀ 𝐴 ∩ ≀ 𝐵)

Proof of Theorem 1cosscnvxrn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 br1cosscnvxrn 37870 . . . . 5 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(𝐴𝐵)𝑦 ↔ (𝑥𝐴𝑦𝑥𝐵𝑦)))
21el2v 3477 . . . 4 (𝑥(𝐴𝐵)𝑦 ↔ (𝑥𝐴𝑦𝑥𝐵𝑦))
32opabbii 5209 . . 3 {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐴𝐵)𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥𝐵𝑦)}
4 inopab 5825 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐵𝑦}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥𝐵𝑦)}
53, 4eqtr4i 2758 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐴𝐵)𝑦} = ({⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐵𝑦})
6 relcoss 37819 . . 3 Rel ≀ (𝐴𝐵)
7 dfrel4v 6188 . . 3 (Rel ≀ (𝐴𝐵) ↔ ≀ (𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐴𝐵)𝑦})
86, 7mpbi 229 . 2 (𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐴𝐵)𝑦}
9 relcoss 37819 . . . 4 Rel ≀ 𝐴
10 dfrel4v 6188 . . . 4 (Rel ≀ 𝐴 ↔ ≀ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦})
119, 10mpbi 229 . . 3 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦}
12 relcoss 37819 . . . 4 Rel ≀ 𝐵
13 dfrel4v 6188 . . . 4 (Rel ≀ 𝐵 ↔ ≀ 𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐵𝑦})
1412, 13mpbi 229 . . 3 𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐵𝑦}
1511, 14ineq12i 4206 . 2 ( ≀ 𝐴 ∩ ≀ 𝐵) = ({⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐵𝑦})
165, 8, 153eqtr4i 2765 1 (𝐴𝐵) = ( ≀ 𝐴 ∩ ≀ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1534  Vcvv 3469  cin 3943   class class class wbr 5142  {copab 5204  ccnv 5671  Rel wrel 5677  cxrn 37569  ccoss 37570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7732
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fo 6548  df-fv 6550  df-1st 7985  df-2nd 7986  df-ec 8718  df-xrn 37767  df-coss 37807
This theorem is referenced by:  disjxrn  38142
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