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

Theorem refssex 23535
Description: Every set in a refinement has a superset in the original cover. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)
Assertion
Ref Expression
refssex ((𝐴Ref𝐵𝑆𝐴) → ∃𝑥𝐵 𝑆𝑥)
Distinct variable groups:   𝑥,𝐵   𝑥,𝑆
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem refssex
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 refrel 23532 . . . . 5 Rel Ref
21brrelex1i 5745 . . . 4 (𝐴Ref𝐵𝐴 ∈ V)
3 eqid 2735 . . . . . 6 𝐴 = 𝐴
4 eqid 2735 . . . . . 6 𝐵 = 𝐵
53, 4isref 23533 . . . . 5 (𝐴 ∈ V → (𝐴Ref𝐵 ↔ ( 𝐵 = 𝐴 ∧ ∀𝑦𝐴𝑥𝐵 𝑦𝑥)))
65simplbda 499 . . . 4 ((𝐴 ∈ V ∧ 𝐴Ref𝐵) → ∀𝑦𝐴𝑥𝐵 𝑦𝑥)
72, 6mpancom 688 . . 3 (𝐴Ref𝐵 → ∀𝑦𝐴𝑥𝐵 𝑦𝑥)
8 sseq1 4021 . . . . 5 (𝑦 = 𝑆 → (𝑦𝑥𝑆𝑥))
98rexbidv 3177 . . . 4 (𝑦 = 𝑆 → (∃𝑥𝐵 𝑦𝑥 ↔ ∃𝑥𝐵 𝑆𝑥))
109rspccv 3619 . . 3 (∀𝑦𝐴𝑥𝐵 𝑦𝑥 → (𝑆𝐴 → ∃𝑥𝐵 𝑆𝑥))
117, 10syl 17 . 2 (𝐴Ref𝐵 → (𝑆𝐴 → ∃𝑥𝐵 𝑆𝑥))
1211imp 406 1 ((𝐴Ref𝐵𝑆𝐴) → ∃𝑥𝐵 𝑆𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059  wrex 3068  Vcvv 3478  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:  reftr  23538  refun0  23539  refssfne  36341
  Copyright terms: Public domain W3C validator