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Theorem rncnvepres 38285
Description: The range of the restricted converse epsilon is the union of the restriction. (Contributed by Peter Mazsa, 11-Feb-2018.) (Revised by Peter Mazsa, 26-Sep-2021.)
Assertion
Ref Expression
rncnvepres ran ( E ↾ 𝐴) = 𝐴

Proof of Theorem rncnvepres
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnopab 5968 . 2 ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥)} = {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦𝑥)}
2 cnvepres 38280 . . 3 ( E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥)}
32rneqi 5951 . 2 ran ( E ↾ 𝐴) = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥)}
4 dfuni2 4914 . . 3 𝐴 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝑥}
5 df-rex 3069 . . . 4 (∃𝑥𝐴 𝑦𝑥 ↔ ∃𝑥(𝑥𝐴𝑦𝑥))
65abbii 2807 . . 3 {𝑦 ∣ ∃𝑥𝐴 𝑦𝑥} = {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦𝑥)}
74, 6eqtri 2763 . 2 𝐴 = {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦𝑥)}
81, 3, 73eqtr4i 2773 1 ran ( E ↾ 𝐴) = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wrex 3068   cuni 4912  {copab 5210   E cep 5588  ccnv 5688  ran crn 5690  cres 5691
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
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-eprel 5589  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701
This theorem is referenced by:  dm1cosscnvepres  38438
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