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Theorem refbas 21294
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 21292 . . 3 Rel Ref
21brrelexi 5148 . 2 (𝐴Ref𝐵𝐴 ∈ V)
3 refbas.1 . . . 4 𝑋 = 𝐴
4 refbas.2 . . . 4 𝑌 = 𝐵
53, 4isref 21293 . . 3 (𝐴 ∈ V → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)))
65simprbda 652 . 2 ((𝐴 ∈ V ∧ 𝐴Ref𝐵) → 𝑌 = 𝑋)
72, 6mpancom 702 1 (𝐴Ref𝐵𝑌 = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  wcel 1988  wral 2909  wrex 2910  Vcvv 3195  wss 3567   cuni 4427   class class class wbr 4644  Refcref 21286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-xp 5110  df-rel 5111  df-ref 21289
This theorem is referenced by:  reftr  21298  refun0  21299  locfinreflem  29881  cmpcref  29891  cmppcmp  29899  refssfne  32328
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