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Theorem dmoprab 5585
 Description: The domain of an operation class abstraction. (Contributed by NM, 17-Mar-1995.) (Revised by David Abernethy, 19-Jun-2012.)
Assertion
Ref Expression
dmoprab dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝜑}
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem dmoprab
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 dfoprab2 5552 . . 3 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
21dmeqi 4536 . 2 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = dom {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
3 dmopab 4546 . 2 dom {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑤 ∣ ∃𝑧𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
4 exrot3 1580 . . . . 5 (∃𝑧𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑦𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
5 19.42v 1786 . . . . . 6 (∃𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧𝜑))
652exbii 1497 . . . . 5 (∃𝑥𝑦𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧𝜑))
74, 6bitri 173 . . . 4 (∃𝑧𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧𝜑))
87abbii 2153 . . 3 {𝑤 ∣ ∃𝑧𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧𝜑)}
9 df-opab 3819 . . 3 {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝜑} = {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧𝜑)}
108, 9eqtr4i 2063 . 2 {𝑤 ∣ ∃𝑧𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝜑}
112, 3, 103eqtri 2064 1 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝜑}
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1243  ∃wex 1381  {cab 2026  ⟨cop 3378  {copab 3817  dom cdm 4345  {coprab 5513 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-dm 4355  df-oprab 5516 This theorem is referenced by:  dmoprabss  5586  reldmoprab  5589  fnoprabg  5602  dmaddpq  6477  dmmulpq  6478
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