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Theorem eqrelriiv 5126
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 5125 . 2 ((Rel 𝐴 ∧ Rel 𝐵) → 𝐴 = 𝐵)
51, 2, 4mp2an 703 1 𝐴 = 𝐵
Colors of variables: wff setvar class
Syntax hints:  wb 194   = wceq 1474  wcel 1976  cop 4130  Rel wrel 5033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-opab 4638  df-xp 5034  df-rel 5035
This theorem is referenced by:  eqbrriv  5127  inopab  5162  difopab  5163  dfres2  5359  restidsing  5364  restidsingOLD  5365  cnvopab  5439  cnv0  5441  cnvdif  5444  difxp  5463  cnvcnvsn  5516  dfco2  5537  coiun  5548  co02  5552  coass  5557  ressn  5574  ovoliunlem1  22994  h2hlm  27027  cnvco1  30709  cnvco2  30710  cnviun  36757  coiun1  36759
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