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Theorem opabdm 29748
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 5321 . 2 dom 𝑅 = {𝑥 ∣ ∃𝑦 𝑥𝑅𝑦}
2 nfopab1 4913 . . . 4 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
32nfeq2 2964 . . 3 𝑥 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
4 nfopab2 4914 . . . . 5 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
54nfeq2 2964 . . . 4 𝑦 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
6 df-br 4845 . . . . 5 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
7 eleq2 2874 . . . . . 6 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (⟨𝑥, 𝑦⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
8 opabid 5177 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
97, 8syl6bb 278 . . . . 5 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (⟨𝑥, 𝑦⟩ ∈ 𝑅𝜑))
106, 9syl5bb 274 . . . 4 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (𝑥𝑅𝑦𝜑))
115, 10exbid 2259 . . 3 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (∃𝑦 𝑥𝑅𝑦 ↔ ∃𝑦𝜑))
123, 11abbid 2924 . 2 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → {𝑥 ∣ ∃𝑦 𝑥𝑅𝑦} = {𝑥 ∣ ∃𝑦𝜑})
131, 12syl5eq 2852 1 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → dom 𝑅 = {𝑥 ∣ ∃𝑦𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1637  wex 1859  wcel 2156  {cab 2792  cop 4376   class class class wbr 4844  {copab 4906  dom cdm 5311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-rab 3105  df-v 3393  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-br 4845  df-opab 4907  df-dm 5321
This theorem is referenced by:  fpwrelmapffslem  29834
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