Theorem eqrelriiv 4826

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 4825	. 2 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → 𝐴 = 𝐵)
5	1, 2, 4	mp2an 426	1 ⊢ 𝐴 = 𝐵

Syntax hints:   ↔ wb 105   = wceq 1398   ∈ wcel 2202  ⟨cop 3676  Rel wrel 4736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-opab 4156  df-xp 4737  df-rel 4738

This theorem is referenced by:  eqbrriv  4827  inopab  4868  difopab  4869  dfres2  5071  restidsing  5075  cnvopab  5145  cnv0  5147  cnvdif  5150  cnvcnvsn  5220  dfco2  5243  coiun  5253  co02  5257  coass  5262  ressn  5284