Theorem reloprab 3022

Description: The operation class abstraction is a relation.

Assertion

Ref	Expression
reloprab	⊢ Rel {⟨⟨x, y⟩, z⟩∣φ}

Distinct variable group(s):   x,z   y,z

Proof of Theorem reloprab

Step	Hyp	Ref	Expression
1		relopab 2494	. 2 ⊢ Rel {⟨w, z⟩∣∃x∃y(w = ⟨x, y⟩ ∧ φ)}
2		dfoprab2 3021	. . 3 ⊢ {⟨⟨x, y⟩, z⟩∣φ} = {⟨w, z⟩∣∃x∃y(w = ⟨x, y⟩ ∧ φ)}
3		releq 2477	. . 3 ⊢ ({⟨⟨x, y⟩, z⟩∣φ} = {⟨w, z⟩∣∃x∃y(w = ⟨x, y⟩ ∧ φ)} → (Rel {⟨⟨x, y⟩, z⟩∣φ} ↔ Rel {⟨w, z⟩∣∃x∃y(w = ⟨x, y⟩ ∧ φ)}))
4	2, 3	ax-mp 6	. 2 ⊢ (Rel {⟨⟨x, y⟩, z⟩∣φ} ↔ Rel {⟨w, z⟩∣∃x∃y(w = ⟨x, y⟩ ∧ φ)})
5	1, 4	mpbir 165	1 ⊢ Rel {⟨⟨x, y⟩, z⟩∣φ}

Syntax hints:   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091  ⟨cop 1810  {copab 2055  Rel wrel 2415  {copab2 3002

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098  df-xp 2424  df-rel 2425  df-oprab 3004

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