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Theorem ssref 21363
 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 2658 . . . 4 (𝑋 = 𝑌𝑌 = 𝑋)
21biimpi 206 . . 3 (𝑋 = 𝑌𝑌 = 𝑋)
323ad2ant3 1104 . 2 ((𝐴𝐶𝐴𝐵𝑋 = 𝑌) → 𝑌 = 𝑋)
4 ssel2 3631 . . . . 5 ((𝐴𝐵𝑥𝐴) → 𝑥𝐵)
543ad2antl2 1244 . . . 4 (((𝐴𝐶𝐴𝐵𝑋 = 𝑌) ∧ 𝑥𝐴) → 𝑥𝐵)
6 ssid 3657 . . . 4 𝑥𝑥
7 sseq2 3660 . . . . 5 (𝑦 = 𝑥 → (𝑥𝑦𝑥𝑥))
87rspcev 3340 . . . 4 ((𝑥𝐵𝑥𝑥) → ∃𝑦𝐵 𝑥𝑦)
95, 6, 8sylancl 695 . . 3 (((𝐴𝐶𝐴𝐵𝑋 = 𝑌) ∧ 𝑥𝐴) → ∃𝑦𝐵 𝑥𝑦)
109ralrimiva 2995 . 2 ((𝐴𝐶𝐴𝐵𝑋 = 𝑌) → ∀𝑥𝐴𝑦𝐵 𝑥𝑦)
11 ssref.1 . . . 4 𝑋 = 𝐴
12 ssref.2 . . . 4 𝑌 = 𝐵
1311, 12isref 21360 . . 3 (𝐴𝐶 → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)))
14133ad2ant1 1102 . 2 ((𝐴𝐶𝐴𝐵𝑋 = 𝑌) → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)))
153, 10, 14mpbir2and 977 1 ((𝐴𝐶𝐴𝐵𝑋 = 𝑌) → 𝐴Ref𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942   ⊆ wss 3607  ∪ cuni 4468   class class class wbr 4685  Refcref 21353 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-ref 21356 This theorem is referenced by:  cmpcref  30045  refssfne  32478
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