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Theorem elec1cnvxrn2 38379
Description: Elementhood in the converse range Cartesian product coset of 𝐴. (Contributed by Peter Mazsa, 11-Jul-2021.)
Assertion
Ref Expression
elec1cnvxrn2 (𝐵𝑉 → (𝐵 ∈ [𝐴](𝑅𝑆) ↔ ∃𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝐵𝑅𝑦𝐵𝑆𝑧)))
Distinct variable groups:   𝑦,𝐴,𝑧   𝑦,𝐵,𝑧   𝑦,𝑅,𝑧   𝑦,𝑆,𝑧   𝑦,𝑉,𝑧

Proof of Theorem elec1cnvxrn2
StepHypRef Expression
1 relcnv 6125 . . 3 Rel (𝑅𝑆)
2 relelec 8791 . . 3 (Rel (𝑅𝑆) → (𝐵 ∈ [𝐴](𝑅𝑆) ↔ 𝐴(𝑅𝑆)𝐵))
31, 2ax-mp 5 . 2 (𝐵 ∈ [𝐴](𝑅𝑆) ↔ 𝐴(𝑅𝑆)𝐵)
4 br1cnvxrn2 38378 . 2 (𝐵𝑉 → (𝐴(𝑅𝑆)𝐵 ↔ ∃𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝐵𝑅𝑦𝐵𝑆𝑧)))
53, 4bitrid 283 1 (𝐵𝑉 → (𝐵 ∈ [𝐴](𝑅𝑆) ↔ ∃𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝐵𝑅𝑦𝐵𝑆𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1086   = wceq 1537  wex 1776  wcel 2106  cop 4637   class class class wbr 5148  ccnv 5688  Rel wrel 5694  [cec 8742  cxrn 38161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-1st 8013  df-2nd 8014  df-ec 8746  df-xrn 38353
This theorem is referenced by:  rnxrn  38380
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