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

Theorem refssex 21224
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 21221 . . . . 5 Rel Ref
21brrelexi 5118 . . . 4 (𝐴Ref𝐵𝐴 ∈ V)
3 eqid 2621 . . . . . 6 𝐴 = 𝐴
4 eqid 2621 . . . . . 6 𝐵 = 𝐵
53, 4isref 21222 . . . . 5 (𝐴 ∈ V → (𝐴Ref𝐵 ↔ ( 𝐵 = 𝐴 ∧ ∀𝑦𝐴𝑥𝐵 𝑦𝑥)))
65simplbda 653 . . . 4 ((𝐴 ∈ V ∧ 𝐴Ref𝐵) → ∀𝑦𝐴𝑥𝐵 𝑦𝑥)
72, 6mpancom 702 . . 3 (𝐴Ref𝐵 → ∀𝑦𝐴𝑥𝐵 𝑦𝑥)
8 sseq1 3605 . . . . 5 (𝑦 = 𝑆 → (𝑦𝑥𝑆𝑥))
98rexbidv 3045 . . . 4 (𝑦 = 𝑆 → (∃𝑥𝐵 𝑦𝑥 ↔ ∃𝑥𝐵 𝑆𝑥))
109rspccv 3292 . . 3 (∀𝑦𝐴𝑥𝐵 𝑦𝑥 → (𝑆𝐴 → ∃𝑥𝐵 𝑆𝑥))
117, 10syl 17 . 2 (𝐴Ref𝐵 → (𝑆𝐴 → ∃𝑥𝐵 𝑆𝑥))
1211imp 445 1 ((𝐴Ref𝐵𝑆𝐴) → ∃𝑥𝐵 𝑆𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  Vcvv 3186  wss 3555   cuni 4402   class class class wbr 4613  Refcref 21215
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 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
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 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-ref 21218
This theorem is referenced by:  reftr  21227  refun0  21228  refssfne  31992
  Copyright terms: Public domain W3C validator