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Theorem eqrelriiv 4633
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
StepHypRef Expression
1 eqreliiv.1 . 2 Rel 𝐴
2 eqreliiv.2 . 2 Rel 𝐵
3 eqreliiv.3 . . 3 (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
43eqrelriv 4632 . 2 ((Rel 𝐴 ∧ Rel 𝐵) → 𝐴 = 𝐵)
51, 2, 4mp2an 422 1 𝐴 = 𝐵
Colors of variables: wff set class
Syntax hints:  wb 104   = wceq 1331  wcel 1480  cop 3530  Rel wrel 4544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-opab 3990  df-xp 4545  df-rel 4546
This theorem is referenced by:  eqbrriv  4634  inopab  4671  difopab  4672  dfres2  4871  cnvopab  4940  cnv0  4942  cnvdif  4945  cnvcnvsn  5015  dfco2  5038  coiun  5048  co02  5052  coass  5057  ressn  5079
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