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Theorem refssex 21124
 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 21121 . . . . 5 Rel Ref
21brrelexi 5082 . . . 4 (𝐴Ref𝐵𝐴 ∈ V)
3 eqid 2610 . . . . . 6 𝐴 = 𝐴
4 eqid 2610 . . . . . 6 𝐵 = 𝐵
53, 4isref 21122 . . . . 5 (𝐴 ∈ V → (𝐴Ref𝐵 ↔ ( 𝐵 = 𝐴 ∧ ∀𝑦𝐴𝑥𝐵 𝑦𝑥)))
65simplbda 652 . . . 4 ((𝐴 ∈ V ∧ 𝐴Ref𝐵) → ∀𝑦𝐴𝑥𝐵 𝑦𝑥)
72, 6mpancom 700 . . 3 (𝐴Ref𝐵 → ∀𝑦𝐴𝑥𝐵 𝑦𝑥)
8 sseq1 3589 . . . . 5 (𝑦 = 𝑆 → (𝑦𝑥𝑆𝑥))
98rexbidv 3034 . . . 4 (𝑦 = 𝑆 → (∃𝑥𝐵 𝑦𝑥 ↔ ∃𝑥𝐵 𝑆𝑥))
109rspccv 3279 . . 3 (∀𝑦𝐴𝑥𝐵 𝑦𝑥 → (𝑆𝐴 → ∃𝑥𝐵 𝑆𝑥))
117, 10syl 17 . 2 (𝐴Ref𝐵 → (𝑆𝐴 → ∃𝑥𝐵 𝑆𝑥))
1211imp 444 1 ((𝐴Ref𝐵𝑆𝐴) → ∃𝑥𝐵 𝑆𝑥)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  ∪ cuni 4372   class class class wbr 4583  Refcref 21115 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-ref 21118 This theorem is referenced by:  reftr  21127  refun0  21128  refssfne  31523
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