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Theorem rnoprab2 5864
 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 5863 . 2 ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} = {𝑧 ∣ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑)}
2 r2ex 2459 . . 3 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))
32abbii 2256 . 2 {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝜑} = {𝑧 ∣ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑)}
41, 3eqtr4i 2164 1 ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝜑}
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   = wceq 1332  ∃wex 1469   ∈ wcel 1481  {cab 2126  ∃wrex 2418  ran crn 4549  {coprab 5784 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-pow 4107  ax-pr 4140 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2692  df-un 3081  df-in 3083  df-ss 3090  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-br 3939  df-opab 3999  df-cnv 4556  df-dm 4558  df-rn 4559  df-oprab 5787 This theorem is referenced by:  rnmpo  5890
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