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Theorem refbas 21535
 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 21533 . . 3 Rel Ref
21brrelexi 5315 . 2 (𝐴Ref𝐵𝐴 ∈ V)
3 refbas.1 . . . 4 𝑋 = 𝐴
4 refbas.2 . . . 4 𝑌 = 𝐵
53, 4isref 21534 . . 3 (𝐴 ∈ V → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)))
65simprbda 654 . 2 ((𝐴 ∈ V ∧ 𝐴Ref𝐵) → 𝑌 = 𝑋)
72, 6mpancom 706 1 (𝐴Ref𝐵𝑌 = 𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1632   ∈ wcel 2139  ∀wral 3050  ∃wrex 3051  Vcvv 3340   ⊆ wss 3715  ∪ cuni 4588   class class class wbr 4804  Refcref 21527 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-ref 21530 This theorem is referenced by:  reftr  21539  refun0  21540  locfinreflem  30237  cmpcref  30247  cmppcmp  30255  refssfne  32680
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