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Theorem rnxrncnvepres 37809
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 37808 . 2 ran (𝑅 ⋉ ( E ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢𝑅𝑥𝑢 E 𝑦)}
2 brcnvep 37672 . . . . . 6 (𝑢 ∈ V → (𝑢 E 𝑦𝑦𝑢))
32elv 3475 . . . . 5 (𝑢 E 𝑦𝑦𝑢)
43anbi1ci 625 . . . 4 ((𝑢𝑅𝑥𝑢 E 𝑦) ↔ (𝑦𝑢𝑢𝑅𝑥))
54rexbii 3089 . . 3 (∃𝑢𝐴 (𝑢𝑅𝑥𝑢 E 𝑦) ↔ ∃𝑢𝐴 (𝑦𝑢𝑢𝑅𝑥))
65opabbii 5209 . 2 {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢𝑅𝑥𝑢 E 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑦𝑢𝑢𝑅𝑥)}
71, 6eqtri 2755 1 ran (𝑅 ⋉ ( E ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑦𝑢𝑢𝑅𝑥)}
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1534  wcel 2099  wrex 3065  Vcvv 3469   class class class wbr 5142  {copab 5204   E cep 5575  ccnv 5671  ran crn 5673  cres 5674  cxrn 37582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-eprel 5576  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 7987  df-2nd 7988  df-ec 8720  df-xrn 37780
This theorem is referenced by: (None)
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