Theorem cleqrel 2483

Description: Extensionality principle for relations. Theorem 3.2(ii) of [Monk1] p. 33.

Assertion

Ref	Expression
cleqrel	⊢ ((Rel A ∧ Rel B) → (A = B ↔ ∀x∀y(⟨x, y⟩ ∈ A ↔ ⟨x, y⟩ ∈ B)))

Distinct variable group(s):   x,y,A   x,B,y

Proof of Theorem cleqrel

Step	Hyp	Ref	Expression
1		relss 2480	. . 3 ⊢ (Rel A → (A ⊆ B ↔ ∀x∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B)))
2		relss 2480	. . 3 ⊢ (Rel B → (B ⊆ A ↔ ∀x∀y(⟨x, y⟩ ∈ B → ⟨x, y⟩ ∈ A)))
3	1, 2	bi2anan9 478	. 2 ⊢ ((Rel A ∧ Rel B) → ((A ⊆ B ∧ B ⊆ A) ↔ (∀x∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) ∧ ∀x∀y(⟨x, y⟩ ∈ B → ⟨x, y⟩ ∈ A))))
4		eqss 1516	. 2 ⊢ (A = B ↔ (A ⊆ B ∧ B ⊆ A))
5		albi 785	. . . 4 ⊢ (∀y(⟨x, y⟩ ∈ A ↔ ⟨x, y⟩ ∈ B) ↔ (∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) ∧ ∀y(⟨x, y⟩ ∈ B → ⟨x, y⟩ ∈ A)))
6	5	bial 695	. . 3 ⊢ (∀x∀y(⟨x, y⟩ ∈ A ↔ ⟨x, y⟩ ∈ B) ↔ ∀x(∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) ∧ ∀y(⟨x, y⟩ ∈ B → ⟨x, y⟩ ∈ A)))
7		19.26 749	. . 3 ⊢ (∀x(∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) ∧ ∀y(⟨x, y⟩ ∈ B → ⟨x, y⟩ ∈ A)) ↔ (∀x∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) ∧ ∀x∀y(⟨x, y⟩ ∈ B → ⟨x, y⟩ ∈ A)))
8	6, 7	bitr 151	. 2 ⊢ (∀x∀y(⟨x, y⟩ ∈ A ↔ ⟨x, y⟩ ∈ B) ↔ (∀x∀y(⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ B) ∧ ∀x∀y(⟨x, y⟩ ∈ B → ⟨x, y⟩ ∈ A)))
9	3, 4, 8	3bitr4g 428	1 ⊢ ((Rel A ∧ Rel B) → (A = B ↔ ∀x∀y(⟨x, y⟩ ∈ A ↔ ⟨x, y⟩ ∈ B)))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672   = wceq 1091   ∈ wcel 1092   ⊆ wss 1487  ⟨cop 1810  Rel wrel 2415

This theorem is referenced by:  cleqreli 2484  reldm0 2550  iss 2599  intirr 2628  cores 2659  dfrel2 2660  coi1 2665  funssres 2698  fn0 2739  fcoi1 2765  fcoi2 2766  fcnvres 2768  fnopabfv 2858  cleqfv 2880  fsn 2895

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

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