Theorem relopabiALT 5847

Description: Alternate proof of relopabi 5846 (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 5229	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
3	1, 2	eqtri 2768	. . 3 ⊢ 𝐴 = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
4		vex 3492	. . . . . . . 8 ⊢ 𝑥 ∈ V
5		vex 3492	. . . . . . . 8 ⊢ 𝑦 ∈ V
6	4, 5	opelvv 5740	. . . . . . 7 ⊢ ⟨𝑥, 𝑦⟩ ∈ (V × V)
7		eleq1 2832	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ (V × V) ↔ ⟨𝑥, 𝑦⟩ ∈ (V × V)))
8	6, 7	mpbiri 258	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 ∈ (V × V))
9	8	adantr 480	. . . . 5 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 ∈ (V × V))
10	9	exlimivv 1931	. . . 4 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 ∈ (V × V))
11	10	abssi 4093	. . 3 ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ⊆ (V × V)
12	3, 11	eqsstri 4043	. 2 ⊢ 𝐴 ⊆ (V × V)
13		df-rel 5707	. 2 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
14	12, 13	mpbir 231	1 ⊢ Rel 𝐴

Syntax hints:   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108  {cab 2717  Vcvv 3488   ⊆ wss 3976  ⟨cop 4654  {copab 5228   × cxp 5698  Rel wrel 5705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-opab 5229  df-xp 5706  df-rel 5707

This theorem is referenced by: (None)