MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  refbas Structured version   Visualization version   GIF version

Theorem refbas 23534
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 23532 . . 3 Rel Ref
21brrelex1i 5745 . 2 (𝐴Ref𝐵𝐴 ∈ V)
3 refbas.1 . . . 4 𝑋 = 𝐴
4 refbas.2 . . . 4 𝑌 = 𝐵
53, 4isref 23533 . . 3 (𝐴 ∈ V → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)))
65simprbda 498 . 2 ((𝐴 ∈ V ∧ 𝐴Ref𝐵) → 𝑌 = 𝑋)
72, 6mpancom 688 1 (𝐴Ref𝐵𝑌 = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  wral 3059  wrex 3068  Vcvv 3478  wss 3963   cuni 4912   class class class wbr 5148  Refcref 23526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-ref 23529
This theorem is referenced by:  reftr  23538  refun0  23539  locfinreflem  33801  cmpcref  33811  cmppcmp  33819  refssfne  36341
  Copyright terms: Public domain W3C validator