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Theorem rnopab 5968
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 5218 . . 3 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
2 nfopab2 5219 . . 3 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
31, 2dfrnf 5964 . 2 ran {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑦 ∣ ∃𝑥 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦}
4 df-br 5149 . . . . 5 (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
5 opabidw 5534 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
64, 5bitri 275 . . . 4 (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦𝜑)
76exbii 1845 . . 3 (∃𝑥 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ∃𝑥𝜑)
87abbii 2807 . 2 {𝑦 ∣ ∃𝑥 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦} = {𝑦 ∣ ∃𝑥𝜑}
93, 8eqtri 2763 1 ran {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑦 ∣ ∃𝑥𝜑}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wex 1776  wcel 2106  {cab 2712  cop 4637   class class class wbr 5148  {copab 5210  ran crn 5690
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-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-cnv 5697  df-dm 5699  df-rn 5700
This theorem is referenced by:  rnopabss  5969  rnopab3  5970  rnmpt  5971  mptpreima  6260  rnoprab  7537  pwfir  9353  marypha2lem4  9476  hartogslem1  9580  rnttrcl  9760  axdc2lem  10486  abrexdomjm  32535  abrexexd  32537  lsmsnorb  33399  satfrnmapom  35355  rnmptsn  37318  abrexdom  37717  rncnvepres  38285  imaopab  42249  tfsconcatrn  43332  modelaxreplem3  44945
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