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Theorem refbas 22215
 Description: A refinement covers the same set. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)
Hypotheses
Ref Expression
refbas.1 𝑋 = 𝐴
refbas.2 𝑌 = 𝐵
Assertion
Ref Expression
refbas (𝐴Ref𝐵𝑌 = 𝑋)

Proof of Theorem refbas
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 refrel 22213 . . 3 Rel Ref
21brrelex1i 5581 . 2 (𝐴Ref𝐵𝐴 ∈ V)
3 refbas.1 . . . 4 𝑋 = 𝐴
4 refbas.2 . . . 4 𝑌 = 𝐵
53, 4isref 22214 . . 3 (𝐴 ∈ V → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)))
65simprbda 502 . 2 ((𝐴 ∈ V ∧ 𝐴Ref𝐵) → 𝑌 = 𝑋)
72, 6mpancom 687 1 (𝐴Ref𝐵𝑌 = 𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071  Vcvv 3409   ⊆ wss 3860  ∪ cuni 4801   class class class wbr 5035  Refcref 22207 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5036  df-opab 5098  df-xp 5533  df-rel 5534  df-ref 22210 This theorem is referenced by:  reftr  22219  refun0  22220  locfinreflem  31315  cmpcref  31325  cmppcmp  31333  refssfne  34122
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