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Theorem 1cosscnvxrn 34657
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 34656 . . . . 5 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(𝐴𝐵)𝑦 ↔ (𝑥𝐴𝑦𝑥𝐵𝑦)))
21el2v 34429 . . . 4 (𝑥(𝐴𝐵)𝑦 ↔ (𝑥𝐴𝑦𝑥𝐵𝑦))
32opabbii 4878 . . 3 {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐴𝐵)𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥𝐵𝑦)}
4 inopab 5423 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐵𝑦}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥𝐵𝑦)}
53, 4eqtr4i 2790 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐴𝐵)𝑦} = ({⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐵𝑦})
6 relcoss 34610 . . 3 Rel ≀ (𝐴𝐵)
7 dfrel4v 5769 . . 3 (Rel ≀ (𝐴𝐵) ↔ ≀ (𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐴𝐵)𝑦})
86, 7mpbi 221 . 2 (𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐴𝐵)𝑦}
9 relcoss 34610 . . . 4 Rel ≀ 𝐴
10 dfrel4v 5769 . . . 4 (Rel ≀ 𝐴 ↔ ≀ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦})
119, 10mpbi 221 . . 3 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦}
12 relcoss 34610 . . . 4 Rel ≀ 𝐵
13 dfrel4v 5769 . . . 4 (Rel ≀ 𝐵 ↔ ≀ 𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐵𝑦})
1412, 13mpbi 221 . . 3 𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐵𝑦}
1511, 14ineq12i 3976 . 2 ( ≀ 𝐴 ∩ ≀ 𝐵) = ({⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐵𝑦})
165, 8, 153eqtr4i 2797 1 (𝐴𝐵) = ( ≀ 𝐴 ∩ ≀ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1652  Vcvv 3350  cin 3733   class class class wbr 4811  {copab 4873  ccnv 5278  Rel wrel 5284  cxrn 34406  ccoss 34407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-fo 6076  df-fv 6078  df-1st 7368  df-2nd 7369  df-ec 7951  df-xrn 34565  df-coss 34601
This theorem is referenced by: (None)
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