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Theorem 1cosscnvxrn 36983
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 36982 . . . . 5 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(𝐴𝐵)𝑦 ↔ (𝑥𝐴𝑦𝑥𝐵𝑦)))
21el2v 3452 . . . 4 (𝑥(𝐴𝐵)𝑦 ↔ (𝑥𝐴𝑦𝑥𝐵𝑦))
32opabbii 5173 . . 3 {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐴𝐵)𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥𝐵𝑦)}
4 inopab 5786 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐵𝑦}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥𝐵𝑦)}
53, 4eqtr4i 2764 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐴𝐵)𝑦} = ({⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐵𝑦})
6 relcoss 36931 . . 3 Rel ≀ (𝐴𝐵)
7 dfrel4v 6143 . . 3 (Rel ≀ (𝐴𝐵) ↔ ≀ (𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐴𝐵)𝑦})
86, 7mpbi 229 . 2 (𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐴𝐵)𝑦}
9 relcoss 36931 . . . 4 Rel ≀ 𝐴
10 dfrel4v 6143 . . . 4 (Rel ≀ 𝐴 ↔ ≀ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦})
119, 10mpbi 229 . . 3 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦}
12 relcoss 36931 . . . 4 Rel ≀ 𝐵
13 dfrel4v 6143 . . . 4 (Rel ≀ 𝐵 ↔ ≀ 𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐵𝑦})
1412, 13mpbi 229 . . 3 𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐵𝑦}
1511, 14ineq12i 4171 . 2 ( ≀ 𝐴 ∩ ≀ 𝐵) = ({⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐵𝑦})
165, 8, 153eqtr4i 2771 1 (𝐴𝐵) = ( ≀ 𝐴 ∩ ≀ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  Vcvv 3444  cin 3910   class class class wbr 5106  {copab 5168  ccnv 5633  Rel wrel 5639  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:  disjxrn  37254
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