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Theorem rnxrncnvepres 36514
Description: Range of a range Cartesian product with a restriction of the converse epsilon relation. (Contributed by Peter Mazsa, 6-Dec-2021.)
Assertion
Ref Expression
rnxrncnvepres ran (𝑅 ⋉ ( E ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑦𝑢𝑢𝑅𝑥)}
Distinct variable groups:   𝑢,𝐴,𝑥,𝑦   𝑢,𝑅,𝑥,𝑦

Proof of Theorem rnxrncnvepres
StepHypRef Expression
1 rnxrnres 36513 . 2 ran (𝑅 ⋉ ( E ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢𝑅𝑥𝑢 E 𝑦)}
2 brcnvep 36392 . . . . . 6 (𝑢 ∈ V → (𝑢 E 𝑦𝑦𝑢))
32elv 3437 . . . . 5 (𝑢 E 𝑦𝑦𝑢)
43anbi1ci 626 . . . 4 ((𝑢𝑅𝑥𝑢 E 𝑦) ↔ (𝑦𝑢𝑢𝑅𝑥))
54rexbii 3180 . . 3 (∃𝑢𝐴 (𝑢𝑅𝑥𝑢 E 𝑦) ↔ ∃𝑢𝐴 (𝑦𝑢𝑢𝑅𝑥))
65opabbii 5146 . 2 {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢𝑅𝑥𝑢 E 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑦𝑢𝑢𝑅𝑥)}
71, 6eqtri 2768 1 ran (𝑅 ⋉ ( E ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑦𝑢𝑢𝑅𝑥)}
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1542  wcel 2110  wrex 3067  Vcvv 3431   class class class wbr 5079  {copab 5141   E cep 5494  ccnv 5588  ran crn 5590  cres 5591  cxrn 36320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356  ax-un 7580
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5489  df-eprel 5495  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6389  df-fun 6433  df-fn 6434  df-f 6435  df-fo 6437  df-fv 6439  df-1st 7818  df-2nd 7819  df-ec 8475  df-xrn 36489
This theorem is referenced by: (None)
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