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Theorem rncnvepres 36027
 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 5799 . 2 ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥)} = {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦𝑥)}
2 cnvepres 36021 . . 3 ( E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥)}
32rneqi 5782 . 2 ran ( E ↾ 𝐴) = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥)}
4 dfuni2 4803 . . 3 𝐴 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝑥}
5 df-rex 3076 . . . 4 (∃𝑥𝐴 𝑦𝑥 ↔ ∃𝑥(𝑥𝐴𝑦𝑥))
65abbii 2823 . . 3 {𝑦 ∣ ∃𝑥𝐴 𝑦𝑥} = {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦𝑥)}
74, 6eqtri 2781 . 2 𝐴 = {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦𝑥)}
81, 3, 73eqtr4i 2791 1 ran ( E ↾ 𝐴) = 𝐴
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2735  ∃wrex 3071  ∪ cuni 4801  {copab 5097   E cep 5437  ◡ccnv 5526  ran crn 5528   ↾ cres 5529 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5036  df-opab 5098  df-eprel 5438  df-xp 5533  df-rel 5534  df-cnv 5535  df-dm 5537  df-rn 5538  df-res 5539 This theorem is referenced by:  dm1cosscnvepres  36162
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