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Theorem ssref 23536
Description: A subcover is a refinement of the original cover. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)
Hypotheses
Ref Expression
ssref.1 𝑋 = 𝐴
ssref.2 𝑌 = 𝐵
Assertion
Ref Expression
ssref ((𝐴𝐶𝐴𝐵𝑋 = 𝑌) → 𝐴Ref𝐵)

Proof of Theorem ssref
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqcom 2742 . . . 4 (𝑋 = 𝑌𝑌 = 𝑋)
21biimpi 216 . . 3 (𝑋 = 𝑌𝑌 = 𝑋)
323ad2ant3 1134 . 2 ((𝐴𝐶𝐴𝐵𝑋 = 𝑌) → 𝑌 = 𝑋)
4 ssel2 3990 . . . . 5 ((𝐴𝐵𝑥𝐴) → 𝑥𝐵)
543ad2antl2 1185 . . . 4 (((𝐴𝐶𝐴𝐵𝑋 = 𝑌) ∧ 𝑥𝐴) → 𝑥𝐵)
6 ssid 4018 . . . 4 𝑥𝑥
7 sseq2 4022 . . . . 5 (𝑦 = 𝑥 → (𝑥𝑦𝑥𝑥))
87rspcev 3622 . . . 4 ((𝑥𝐵𝑥𝑥) → ∃𝑦𝐵 𝑥𝑦)
95, 6, 8sylancl 586 . . 3 (((𝐴𝐶𝐴𝐵𝑋 = 𝑌) ∧ 𝑥𝐴) → ∃𝑦𝐵 𝑥𝑦)
109ralrimiva 3144 . 2 ((𝐴𝐶𝐴𝐵𝑋 = 𝑌) → ∀𝑥𝐴𝑦𝐵 𝑥𝑦)
11 ssref.1 . . . 4 𝑋 = 𝐴
12 ssref.2 . . . 4 𝑌 = 𝐵
1311, 12isref 23533 . . 3 (𝐴𝐶 → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)))
14133ad2ant1 1132 . 2 ((𝐴𝐶𝐴𝐵𝑋 = 𝑌) → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)))
153, 10, 14mpbir2and 713 1 ((𝐴𝐶𝐴𝐵𝑋 = 𝑌) → 𝐴Ref𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wrex 3068  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:  cmpcref  33811  refssfne  36341
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