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Theorem rnoprab 5820
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 5784 . . 3 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
21rneqi 4735 . 2 ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = ran {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
3 rnopab 4754 . 2 ran {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑧 ∣ ∃𝑤𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
4 exrot3 1651 . . . 4 (∃𝑤𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑦𝑤(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
5 vex 2661 . . . . . . . 8 𝑥 ∈ V
6 vex 2661 . . . . . . . 8 𝑦 ∈ V
75, 6opex 4119 . . . . . . 7 𝑥, 𝑦⟩ ∈ V
87isseti 2666 . . . . . 6 𝑤 𝑤 = ⟨𝑥, 𝑦
9 19.41v 1856 . . . . . 6 (∃𝑤(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (∃𝑤 𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
108, 9mpbiran 907 . . . . 5 (∃𝑤(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜑)
11102exbii 1568 . . . 4 (∃𝑥𝑦𝑤(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑦𝜑)
124, 11bitri 183 . . 3 (∃𝑤𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑦𝜑)
1312abbii 2231 . 2 {𝑧 ∣ ∃𝑤𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑧 ∣ ∃𝑥𝑦𝜑}
142, 3, 133eqtri 2140 1 ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦𝜑}
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1314  wex 1451  {cab 2101  cop 3498  {copab 3956  ran crn 4508  {coprab 5741
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-cnv 4515  df-dm 4517  df-rn 4518  df-oprab 5744
This theorem is referenced by:  rnoprab2  5821
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