Theorem eqrelriv 3246

Description: Inference from extensionality principle for relations.

Hypotheses

Ref	Expression
eqreli.1	⊢ Rel A
eqreli.2	⊢ Rel B
eqreli.3	⊢ (⟨x, y⟩ ∈ A ↔ ⟨x, y⟩ ∈ B)

Assertion

Ref	Expression
eqrelriv	⊢ A = B

Distinct variable groups:   x,y,A   x,B,y

Proof of Theorem eqrelriv

Step	Hyp	Ref	Expression
1		eqreli.1	. . 3 ⊢ Rel A
2		eqreli.2	. . 3 ⊢ Rel B
3		eqrel 3245	. . 3 ⊢ ((Rel A ⋀ Rel B) → (A = B ↔ ∀x∀y(⟨x, y⟩ ∈ A ↔ ⟨x, y⟩ ∈ B)))
4	1, 2, 3	mp2an 696	. 2 ⊢ (A = B ↔ ∀x∀y(⟨x, y⟩ ∈ A ↔ ⟨x, y⟩ ∈ B))
5		eqreli.3	. . 3 ⊢ (⟨x, y⟩ ∈ A ↔ ⟨x, y⟩ ∈ B)
6	5	ax-gen 961	. 2 ⊢ ∀y(⟨x, y⟩ ∈ A ↔ ⟨x, y⟩ ∈ B)
7	4, 6	mpgbir 986	1 ⊢ A = B

Syntax hints:   ↔ wb 146  ∀wal 952   = wceq 954   ∈ wcel 956  ⟨cop 2407  Rel wrel 3170

This theorem is referenced by:  eqbrriv 3247  inopab 3263  inxp 3264  cnvopab 3437  cnv0 3438  cnvi 3439  cnvsn 3441  cnvun 3447  cnvin 3448  cnvxp 3456  co02 3500  coass 3504  sbthcl 4445

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-opab 2662  df-xp 3179  df-rel 3180

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