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Theorem opabrn 30380
 Description: Range of an ordered-pair class abstraction. (Contributed by Thierry Arnoux, 31-Aug-2017.)
Assertion
Ref Expression
opabrn (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ran 𝑅 = {𝑦 ∣ ∃𝑥𝜑})
Distinct variable group:   𝑥,𝑦,𝑅
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem opabrn
StepHypRef Expression
1 dfrn2 5727 . 2 ran 𝑅 = {𝑦 ∣ ∃𝑥 𝑥𝑅𝑦}
2 nfopab2 5103 . . . 4 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
32nfeq2 2975 . . 3 𝑦 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
4 nfopab1 5102 . . . . 5 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
54nfeq2 2975 . . . 4 𝑥 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
6 df-br 5034 . . . . 5 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
7 eleq2 2881 . . . . . 6 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (⟨𝑥, 𝑦⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
8 opabidw 5380 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
97, 8syl6bb 290 . . . . 5 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (⟨𝑥, 𝑦⟩ ∈ 𝑅𝜑))
106, 9syl5bb 286 . . . 4 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (𝑥𝑅𝑦𝜑))
115, 10exbid 2224 . . 3 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (∃𝑥 𝑥𝑅𝑦 ↔ ∃𝑥𝜑))
123, 11abbid 2867 . 2 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → {𝑦 ∣ ∃𝑥 𝑥𝑅𝑦} = {𝑦 ∣ ∃𝑥𝜑})
131, 12syl5eq 2848 1 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ran 𝑅 = {𝑦 ∣ ∃𝑥𝜑})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538  ∃wex 1781   ∈ wcel 2112  {cab 2779  ⟨cop 4534   class class class wbr 5033  {copab 5095  ran crn 5524 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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-cnv 5531  df-dm 5533  df-rn 5534 This theorem is referenced by:  fpwrelmapffslem  30498
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