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Theorem opabdm 32108
Description: Domain of an ordered-pair class abstraction. (Contributed by Thierry Arnoux, 31-Aug-2017.)
Assertion
Ref Expression
opabdm (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → dom 𝑅 = {𝑥 ∣ ∃𝑦𝜑})
Distinct variable group:   𝑥,𝑦,𝑅
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem opabdm
StepHypRef Expression
1 df-dm 5686 . 2 dom 𝑅 = {𝑥 ∣ ∃𝑦 𝑥𝑅𝑦}
2 nfopab1 5218 . . . 4 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
32nfeq2 2919 . . 3 𝑥 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
4 nfopab2 5219 . . . . 5 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
54nfeq2 2919 . . . 4 𝑦 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
6 df-br 5149 . . . . 5 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
7 eleq2 2821 . . . . . 6 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (⟨𝑥, 𝑦⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
8 opabidw 5524 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
97, 8bitrdi 287 . . . . 5 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (⟨𝑥, 𝑦⟩ ∈ 𝑅𝜑))
106, 9bitrid 283 . . . 4 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (𝑥𝑅𝑦𝜑))
115, 10exbid 2215 . . 3 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (∃𝑦 𝑥𝑅𝑦 ↔ ∃𝑦𝜑))
123, 11abbid 2802 . 2 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → {𝑥 ∣ ∃𝑦 𝑥𝑅𝑦} = {𝑥 ∣ ∃𝑦𝜑})
131, 12eqtrid 2783 1 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → dom 𝑅 = {𝑥 ∣ ∃𝑦𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wex 1780  wcel 2105  {cab 2708  cop 4634   class class class wbr 5148  {copab 5210  dom cdm 5676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-dm 5686
This theorem is referenced by:  fpwrelmapffslem  32225
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