Theorem eqrelriiv 5776

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 5775	. 2 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → 𝐴 = 𝐵)
5	1, 2, 4	mp2an 704	1 ⊢ 𝐴 = 𝐵

Syntax hints:   ↔ wb 209   = wceq 1570   ∈ wcel 2143  ⟨cop 4595  Rel wrel 5666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-ss 3922  df-opab 5174  df-xp 5667  df-rel 5668

This theorem is referenced by:  eqbrriv  5777  inopab  5816  difopab  5817  inxp  5818  dfres2  6043  restidsing  6055  cnvopab  6137  cnvdif  6140  difxp  6161  cnvcnvsn  6220  dfco2  6246  coiun  6258  co02  6262  coass  6267  ressn  6286  ovoliunlem1  25661  h2hlm  31332  cnvco1  36251  cnvco2  36252  inxprnres  38967  cnviun  44396  coiun1  44398  coxp  49631