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Theorem relopabiALT 5836
Description: Alternate proof of relopabi 5835 (shorter but uses more axioms). (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 5211 . . . 4 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
31, 2eqtri 2763 . . 3 𝐴 = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
4 vex 3482 . . . . . . . 8 𝑥 ∈ V
5 vex 3482 . . . . . . . 8 𝑦 ∈ V
64, 5opelvv 5729 . . . . . . 7 𝑥, 𝑦⟩ ∈ (V × V)
7 eleq1 2827 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ (V × V) ↔ ⟨𝑥, 𝑦⟩ ∈ (V × V)))
86, 7mpbiri 258 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 ∈ (V × V))
98adantr 480 . . . . 5 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 ∈ (V × V))
109exlimivv 1930 . . . 4 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 ∈ (V × V))
1110abssi 4080 . . 3 {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ⊆ (V × V)
123, 11eqsstri 4030 . 2 𝐴 ⊆ (V × V)
13 df-rel 5696 . 2 (Rel 𝐴𝐴 ⊆ (V × V))
1412, 13mpbir 231 1 Rel 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1537  wex 1776  wcel 2106  {cab 2712  Vcvv 3478  wss 3963  cop 4637  {copab 5210   × cxp 5687  Rel wrel 5694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-opab 5211  df-xp 5695  df-rel 5696
This theorem is referenced by: (None)
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