Theorem eqrelriiv 5657

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 5656	. 2 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → 𝐴 = 𝐵)
5	1, 2, 4	mp2an 690	1 ⊢ 𝐴 = 𝐵

Syntax hints:   ↔ wb 208   = wceq 1533   ∈ wcel 2110  ⟨cop 4566  Rel wrel 5554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-in 3942  df-ss 3951  df-opab 5121  df-xp 5555  df-rel 5556

This theorem is referenced by:  eqbrriv  5658  inopab  5695  difopab  5696  dfres2  5903  restidsing  5916  cnvopab  5991  cnvdif  5996  difxp  6015  cnvcnvsn  6070  dfco2  6092  coiun  6103  co02  6107  coass  6112  ressn  6130  ovoliunlem1  24097  h2hlm  28751  cnvco1  32990  cnvco2  32991  inxprnres  35543  cnviun  39988  coiun1  39990