Theorem eqrelriiv 5770

Description: Inference from extensionality principle for relations. (Contributed by NM, 17-Mar-1995.)

Hypotheses

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

Assertion

Ref	Expression
eqrelriiv	⊢ 𝐴 = 𝐵

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

Proof of Theorem eqrelriiv

Step	Hyp	Ref	Expression
1		eqreliiv.1	. 2 ⊢ Rel 𝐴
2		eqreliiv.2	. 2 ⊢ Rel 𝐵
3		eqreliiv.3	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
4	3	eqrelriv 5769	. 2 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → 𝐴 = 𝐵)
5	1, 2, 4	mp2an 705	1 ⊢ 𝐴 = 𝐵

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569   ∈ wcel 2146  ⟨cop 4837  Rel wrel 5659

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3455  df-ss 3900  df-opab 5170  df-xp 5660  df-rel 5661

This theorem is used by:  eqbrriv  5771  inopab  5811  difopab  5812  inxp  5813  dfres2  6038  restidsing  6050  cnvopab  6136  cnvdif  6139  difxp  6160  cnvcnvsn  6226  dfco2  6252  coiun  6264  co02  6268  coass  6273  ressn  6294  ovoliunlem1  25835  h2hlm  31594  cnvco1  36651  cnvco2  36652  inxprnres  39370  cnviun  44790  coiun1  44792  coxp  50083