MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rnopab Structured version   Visualization version   GIF version

Theorem rnopab 5819
Description: The range of a class of ordered pairs. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)
Assertion
Ref Expression
rnopab ran {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑦 ∣ ∃𝑥𝜑}
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem rnopab
StepHypRef Expression
1 nfopab1 5126 . . 3 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
2 nfopab2 5127 . . 3 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
31, 2dfrnf 5813 . 2 ran {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑦 ∣ ∃𝑥 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦}
4 df-br 5058 . . . . 5 (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
5 opabidw 5403 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
64, 5bitri 276 . . . 4 (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦𝜑)
76exbii 1839 . . 3 (∃𝑥 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ∃𝑥𝜑)
87abbii 2883 . 2 {𝑦 ∣ ∃𝑥 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦} = {𝑦 ∣ ∃𝑥𝜑}
93, 8eqtri 2841 1 ran {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑦 ∣ ∃𝑥𝜑}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wex 1771  wcel 2105  {cab 2796  cop 4563   class class class wbr 5057  {copab 5119  ran crn 5549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-cnv 5556  df-dm 5558  df-rn 5559
This theorem is referenced by:  rnmpt  5820  mptpreima  6085  rnoprab  7246  marypha2lem4  8890  hartogslem1  8994  axdc2lem  9858  abrexdomjm  30194  abrexexd  30196  satfrnmapom  32514  rnmptsn  34498  abrexdom  34886  rncnvepres  35442  imaopab  38997
  Copyright terms: Public domain W3C validator