Theorem relopabiALT 5808

Description: Alternate proof of relopabi 5807 (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.

Step	Hyp	Ref	Expression
1		relopabi.1	. . . 4 ⊢ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2		df-opab 5175	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
3	1, 2	eqtri 2792	. . 3 ⊢ 𝐴 = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
4		vex 3467	. . . . . . . 8 ⊢ 𝑥 ∈ V
5		vex 3467	. . . . . . . 8 ⊢ 𝑦 ∈ V
6	4, 5	opelvv 5699	. . . . . . 7 ⊢ ⟨𝑥, 𝑦⟩ ∈ (V × V)
7		eleq1 2857	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ (V × V) ↔ ⟨𝑥, 𝑦⟩ ∈ (V × V)))
8	6, 7	mpbiri 261	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 ∈ (V × V))
9	8	adantr 485	. . . . 5 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 ∈ (V × V))
10	9	exlimivv 1959	. . . 4 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 ∈ (V × V))
11	10	abssi 4030	. . 3 ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ⊆ (V × V)
12	3, 11	eqsstri 3991	. 2 ⊢ 𝐴 ⊆ (V × V)
13		df-rel 5666	. 2 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
14	12, 13	mpbir 234	1 ⊢ Rel 𝐴

Syntax hints:   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149  {cab 2747  Vcvv 3463   ⊆ wss 3913  ⟨cop 4597  {copab 5174   × cxp 5657  Rel wrel 5664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-opab 5175  df-xp 5665  df-rel 5666

This theorem is referenced by: (None)