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Theorem rnoprab 7538
Description: The range of an operation class abstraction. (Contributed by NM, 30-Aug-2004.) (Revised by David Abernethy, 19-Apr-2013.)
Assertion
Ref Expression
rnoprab ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦𝜑}
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem rnoprab
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 dfoprab2 7491 . . 3 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
21rneqi 5948 . 2 ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = ran {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
3 rnopab 5965 . 2 ran {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑧 ∣ ∃𝑤𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
4 exrot3 2165 . . . 4 (∃𝑤𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑦𝑤(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
5 opex 5469 . . . . . . 7 𝑥, 𝑦⟩ ∈ V
65isseti 3498 . . . . . 6 𝑤 𝑤 = ⟨𝑥, 𝑦
7 19.41v 1949 . . . . . 6 (∃𝑤(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (∃𝑤 𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
86, 7mpbiran 709 . . . . 5 (∃𝑤(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜑)
982exbii 1849 . . . 4 (∃𝑥𝑦𝑤(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑦𝜑)
104, 9bitri 275 . . 3 (∃𝑤𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑦𝜑)
1110abbii 2809 . 2 {𝑧 ∣ ∃𝑤𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑧 ∣ ∃𝑥𝑦𝜑}
122, 3, 113eqtri 2769 1 ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦𝜑}
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  wex 1779  {cab 2714  cop 4632  {copab 5205  ran crn 5686  {coprab 7432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-cnv 5693  df-dm 5695  df-rn 5696  df-oprab 7435
This theorem is referenced by:  rnoprab2  7539  elrnmpores  7571  ellines  36153
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