Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  1cosscnvxrn Structured version   Visualization version   GIF version

Theorem 1cosscnvxrn 38002
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 38001 . . . . 5 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(𝐴𝐵)𝑦 ↔ (𝑥𝐴𝑦𝑥𝐵𝑦)))
21el2v 3471 . . . 4 (𝑥(𝐴𝐵)𝑦 ↔ (𝑥𝐴𝑦𝑥𝐵𝑦))
32opabbii 5210 . . 3 {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐴𝐵)𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥𝐵𝑦)}
4 inopab 5825 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐵𝑦}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥𝐵𝑦)}
53, 4eqtr4i 2756 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐴𝐵)𝑦} = ({⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐵𝑦})
6 relcoss 37950 . . 3 Rel ≀ (𝐴𝐵)
7 dfrel4v 6189 . . 3 (Rel ≀ (𝐴𝐵) ↔ ≀ (𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐴𝐵)𝑦})
86, 7mpbi 229 . 2 (𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐴𝐵)𝑦}
9 relcoss 37950 . . . 4 Rel ≀ 𝐴
10 dfrel4v 6189 . . . 4 (Rel ≀ 𝐴 ↔ ≀ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦})
119, 10mpbi 229 . . 3 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦}
12 relcoss 37950 . . . 4 Rel ≀ 𝐵
13 dfrel4v 6189 . . . 4 (Rel ≀ 𝐵 ↔ ≀ 𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐵𝑦})
1412, 13mpbi 229 . . 3 𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐵𝑦}
1511, 14ineq12i 4204 . 2 ( ≀ 𝐴 ∩ ≀ 𝐵) = ({⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐵𝑦})
165, 8, 153eqtr4i 2763 1 (𝐴𝐵) = ( ≀ 𝐴 ∩ ≀ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1533  Vcvv 3463  cin 3939   class class class wbr 5143  {copab 5205  ccnv 5671  Rel wrel 5677  cxrn 37703  ccoss 37704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  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 7989  df-2nd 7990  df-ec 8723  df-xrn 37898  df-coss 37938
This theorem is referenced by:  disjxrn  38273
  Copyright terms: Public domain W3C validator