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Theorem rnoprab2 6642
 Description: The range of a restricted operation class abstraction. (Contributed by Scott Fenton, 21-Mar-2012.)
Assertion
Ref Expression
rnoprab2 ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝜑}
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐴(𝑥,𝑧)   𝐵(𝑥,𝑦,𝑧)

Proof of Theorem rnoprab2
StepHypRef Expression
1 rnoprab 6641 . 2 ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} = {𝑧 ∣ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑)}
2 r2ex 3043 . . 3 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))
32abbii 2726 . 2 {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝜑} = {𝑧 ∣ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑)}
41, 3eqtr4i 2635 1 ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝜑}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  ∃wrex 2897  ran crn 5039  {coprab 6550 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-cnv 5046  df-dm 5048  df-rn 5049  df-oprab 6553 This theorem is referenced by:  rnmpt2  6668
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