Theorem eqrelriiv 4754

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 4753	. 2 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → 𝐴 = 𝐵)
5	1, 2, 4	mp2an 426	1 ⊢ 𝐴 = 𝐵

Syntax hints:   ↔ wb 105   = wceq 1364   ∈ wcel 2164  ⟨cop 3622  Rel wrel 4665

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-opab 4092  df-xp 4666  df-rel 4667

This theorem is referenced by:  eqbrriv  4755  inopab  4795  difopab  4796  dfres2  4995  restidsing  4999  cnvopab  5068  cnv0  5070  cnvdif  5073  cnvcnvsn  5143  dfco2  5166  coiun  5176  co02  5180  coass  5185  ressn  5207