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Theorem 1cosscnvxrn 35259
 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 35258 . . . . 5 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(𝐴𝐵)𝑦 ↔ (𝑥𝐴𝑦𝑥𝐵𝑦)))
21el2v 3443 . . . 4 (𝑥(𝐴𝐵)𝑦 ↔ (𝑥𝐴𝑦𝑥𝐵𝑦))
32opabbii 5031 . . 3 {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐴𝐵)𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥𝐵𝑦)}
4 inopab 5590 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐵𝑦}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥𝐵𝑦)}
53, 4eqtr4i 2821 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐴𝐵)𝑦} = ({⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐵𝑦})
6 relcoss 35212 . . 3 Rel ≀ (𝐴𝐵)
7 dfrel4v 5926 . . 3 (Rel ≀ (𝐴𝐵) ↔ ≀ (𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐴𝐵)𝑦})
86, 7mpbi 231 . 2 (𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐴𝐵)𝑦}
9 relcoss 35212 . . . 4 Rel ≀ 𝐴
10 dfrel4v 5926 . . . 4 (Rel ≀ 𝐴 ↔ ≀ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦})
119, 10mpbi 231 . . 3 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦}
12 relcoss 35212 . . . 4 Rel ≀ 𝐵
13 dfrel4v 5926 . . . 4 (Rel ≀ 𝐵 ↔ ≀ 𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐵𝑦})
1412, 13mpbi 231 . . 3 𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐵𝑦}
1511, 14ineq12i 4109 . 2 ( ≀ 𝐴 ∩ ≀ 𝐵) = ({⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐵𝑦})
165, 8, 153eqtr4i 2828 1 (𝐴𝐵) = ( ≀ 𝐴 ∩ ≀ 𝐵)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 207   ∧ wa 396   = wceq 1522  Vcvv 3436   ∩ cin 3860   class class class wbr 4964  {copab 5026  ◡ccnv 5445  Rel wrel 5451   ⋉ cxrn 34997   ≀ ccoss 34998 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-sep 5097  ax-nul 5104  ax-pow 5160  ax-pr 5224  ax-un 7322 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ne 2984  df-ral 3109  df-rex 3110  df-rab 3113  df-v 3438  df-sbc 3708  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-nul 4214  df-if 4384  df-sn 4475  df-pr 4477  df-op 4481  df-uni 4748  df-br 4965  df-opab 5027  df-mpt 5044  df-id 5351  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-rn 5457  df-res 5458  df-ima 5459  df-iota 6192  df-fun 6230  df-fn 6231  df-f 6232  df-fo 6234  df-fv 6236  df-1st 7548  df-2nd 7549  df-ec 8144  df-xrn 35167  df-coss 35203 This theorem is referenced by:  disjxrn  35521
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