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Theorem rnoprab2 7253
 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 7252 . 2 ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} = {𝑧 ∣ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑)}
2 r2ex 3228 . . 3 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))
32abbii 2824 . 2 {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝜑} = {𝑧 ∣ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑)}
41, 3eqtr4i 2785 1 ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝜑}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 400   = wceq 1539  ∃wex 1782   ∈ wcel 2112  {cab 2736  ∃wrex 3072  ran crn 5526  {coprab 7152 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-v 3412  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-br 5034  df-opab 5096  df-cnv 5533  df-dm 5535  df-rn 5536  df-oprab 7155 This theorem is referenced by:  rnmpo  7280
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