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Theorem ssref 22123
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 2831 . . . 4 (𝑋 = 𝑌𝑌 = 𝑋)
21biimpi 219 . . 3 (𝑋 = 𝑌𝑌 = 𝑋)
323ad2ant3 1132 . 2 ((𝐴𝐶𝐴𝐵𝑋 = 𝑌) → 𝑌 = 𝑋)
4 ssel2 3948 . . . . 5 ((𝐴𝐵𝑥𝐴) → 𝑥𝐵)
543ad2antl2 1183 . . . 4 (((𝐴𝐶𝐴𝐵𝑋 = 𝑌) ∧ 𝑥𝐴) → 𝑥𝐵)
6 ssid 3975 . . . 4 𝑥𝑥
7 sseq2 3979 . . . . 5 (𝑦 = 𝑥 → (𝑥𝑦𝑥𝑥))
87rspcev 3609 . . . 4 ((𝑥𝐵𝑥𝑥) → ∃𝑦𝐵 𝑥𝑦)
95, 6, 8sylancl 589 . . 3 (((𝐴𝐶𝐴𝐵𝑋 = 𝑌) ∧ 𝑥𝐴) → ∃𝑦𝐵 𝑥𝑦)
109ralrimiva 3177 . 2 ((𝐴𝐶𝐴𝐵𝑋 = 𝑌) → ∀𝑥𝐴𝑦𝐵 𝑥𝑦)
11 ssref.1 . . . 4 𝑋 = 𝐴
12 ssref.2 . . . 4 𝑌 = 𝐵
1311, 12isref 22120 . . 3 (𝐴𝐶 → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)))
14133ad2ant1 1130 . 2 ((𝐴𝐶𝐴𝐵𝑋 = 𝑌) → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)))
153, 10, 14mpbir2and 712 1 ((𝐴𝐶𝐴𝐵𝑋 = 𝑌) → 𝐴Ref𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3133  wrex 3134  wss 3919   cuni 4824   class class class wbr 5052  Refcref 22113
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-xp 5548  df-rel 5549  df-ref 22116
This theorem is referenced by:  cmpcref  31177  refssfne  33766
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