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Theorem relopabiALT 5402
Description: Alternate proof of relopabi 5401 (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 4865 . . . 4 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
31, 2eqtri 2782 . . 3 𝐴 = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
4 vex 3343 . . . . . . . 8 𝑥 ∈ V
5 vex 3343 . . . . . . . 8 𝑦 ∈ V
64, 5opelvv 5323 . . . . . . 7 𝑥, 𝑦⟩ ∈ (V × V)
7 eleq1 2827 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ (V × V) ↔ ⟨𝑥, 𝑦⟩ ∈ (V × V)))
86, 7mpbiri 248 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 ∈ (V × V))
98adantr 472 . . . . 5 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 ∈ (V × V))
109exlimivv 2009 . . . 4 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 ∈ (V × V))
1110abssi 3818 . . 3 {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ⊆ (V × V)
123, 11eqsstri 3776 . 2 𝐴 ⊆ (V × V)
13 df-rel 5273 . 2 (Rel 𝐴𝐴 ⊆ (V × V))
1412, 13mpbir 221 1 Rel 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1632  wex 1853  wcel 2139  {cab 2746  Vcvv 3340  wss 3715  cop 4327  {copab 4864   × cxp 5264  Rel wrel 5271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-opab 4865  df-xp 5272  df-rel 5273
This theorem is referenced by: (None)
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