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Theorem relopabiALT 5168
 Description: Alternate proof of relopabi 5167. (Contributed by Mario Carneiro, 21-Dec-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
relopabi.1 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Assertion
Ref Expression
relopabiALT Rel 𝐴

Proof of Theorem relopabiALT
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 relopabi.1 . . . 4 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2 df-opab 4644 . . . 4 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
31, 2eqtri 2632 . . 3 𝐴 = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
4 vex 3176 . . . . . . . 8 𝑥 ∈ V
5 vex 3176 . . . . . . . 8 𝑦 ∈ V
64, 5opelvv 5088 . . . . . . 7 𝑥, 𝑦⟩ ∈ (V × V)
7 eleq1 2676 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ (V × V) ↔ ⟨𝑥, 𝑦⟩ ∈ (V × V)))
86, 7mpbiri 247 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 ∈ (V × V))
98adantr 480 . . . . 5 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 ∈ (V × V))
109exlimivv 1847 . . . 4 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 ∈ (V × V))
1110abssi 3640 . . 3 {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ⊆ (V × V)
123, 11eqsstri 3598 . 2 𝐴 ⊆ (V × V)
13 df-rel 5045 . 2 (Rel 𝐴𝐴 ⊆ (V × V))
1412, 13mpbir 220 1 Rel 𝐴
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  Vcvv 3173   ⊆ wss 3540  ⟨cop 4131  {copab 4642   × cxp 5036  Rel wrel 5043 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644  df-xp 5044  df-rel 5045 This theorem is referenced by: (None)
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