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Theorem ssref 22886
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 2740 . . . 4 (𝑋 = 𝑌𝑌 = 𝑋)
21biimpi 215 . . 3 (𝑋 = 𝑌𝑌 = 𝑋)
323ad2ant3 1136 . 2 ((𝐴𝐶𝐴𝐵𝑋 = 𝑌) → 𝑌 = 𝑋)
4 ssel2 3943 . . . . 5 ((𝐴𝐵𝑥𝐴) → 𝑥𝐵)
543ad2antl2 1187 . . . 4 (((𝐴𝐶𝐴𝐵𝑋 = 𝑌) ∧ 𝑥𝐴) → 𝑥𝐵)
6 ssid 3970 . . . 4 𝑥𝑥
7 sseq2 3974 . . . . 5 (𝑦 = 𝑥 → (𝑥𝑦𝑥𝑥))
87rspcev 3583 . . . 4 ((𝑥𝐵𝑥𝑥) → ∃𝑦𝐵 𝑥𝑦)
95, 6, 8sylancl 587 . . 3 (((𝐴𝐶𝐴𝐵𝑋 = 𝑌) ∧ 𝑥𝐴) → ∃𝑦𝐵 𝑥𝑦)
109ralrimiva 3140 . 2 ((𝐴𝐶𝐴𝐵𝑋 = 𝑌) → ∀𝑥𝐴𝑦𝐵 𝑥𝑦)
11 ssref.1 . . . 4 𝑋 = 𝐴
12 ssref.2 . . . 4 𝑌 = 𝐵
1311, 12isref 22883 . . 3 (𝐴𝐶 → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)))
14133ad2ant1 1134 . 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 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3061  wrex 3070  wss 3914   cuni 4869   class class class wbr 5109  Refcref 22876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-xp 5643  df-rel 5644  df-ref 22879
This theorem is referenced by:  cmpcref  32495  refssfne  34883
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