MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssref Structured version   Visualization version   GIF version

Theorem ssref 21220
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 2633 . . . 4 (𝑋 = 𝑌𝑌 = 𝑋)
21biimpi 206 . . 3 (𝑋 = 𝑌𝑌 = 𝑋)
323ad2ant3 1082 . 2 ((𝐴𝐶𝐴𝐵𝑋 = 𝑌) → 𝑌 = 𝑋)
4 ssel2 3583 . . . . 5 ((𝐴𝐵𝑥𝐴) → 𝑥𝐵)
543ad2antl2 1222 . . . 4 (((𝐴𝐶𝐴𝐵𝑋 = 𝑌) ∧ 𝑥𝐴) → 𝑥𝐵)
6 ssid 3608 . . . 4 𝑥𝑥
7 sseq2 3611 . . . . 5 (𝑦 = 𝑥 → (𝑥𝑦𝑥𝑥))
87rspcev 3300 . . . 4 ((𝑥𝐵𝑥𝑥) → ∃𝑦𝐵 𝑥𝑦)
95, 6, 8sylancl 693 . . 3 (((𝐴𝐶𝐴𝐵𝑋 = 𝑌) ∧ 𝑥𝐴) → ∃𝑦𝐵 𝑥𝑦)
109ralrimiva 2965 . 2 ((𝐴𝐶𝐴𝐵𝑋 = 𝑌) → ∀𝑥𝐴𝑦𝐵 𝑥𝑦)
11 ssref.1 . . . 4 𝑋 = 𝐴
12 ssref.2 . . . 4 𝑌 = 𝐵
1311, 12isref 21217 . . 3 (𝐴𝐶 → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)))
14133ad2ant1 1080 . 2 ((𝐴𝐶𝐴𝐵𝑋 = 𝑌) → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)))
153, 10, 14mpbir2and 956 1 ((𝐴𝐶𝐴𝐵𝑋 = 𝑌) → 𝐴Ref𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1992  wral 2912  wrex 2913  wss 3560   cuni 4407   class class class wbr 4618  Refcref 21210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-xp 5085  df-rel 5086  df-ref 21213
This theorem is referenced by:  cmpcref  29691  refssfne  31968
  Copyright terms: Public domain W3C validator