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Theorem refssex 22047
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 22044 . . . . 5 Rel Ref
21brrelex1i 5601 . . . 4 (𝐴Ref𝐵𝐴 ∈ V)
3 eqid 2818 . . . . . 6 𝐴 = 𝐴
4 eqid 2818 . . . . . 6 𝐵 = 𝐵
53, 4isref 22045 . . . . 5 (𝐴 ∈ V → (𝐴Ref𝐵 ↔ ( 𝐵 = 𝐴 ∧ ∀𝑦𝐴𝑥𝐵 𝑦𝑥)))
65simplbda 500 . . . 4 ((𝐴 ∈ V ∧ 𝐴Ref𝐵) → ∀𝑦𝐴𝑥𝐵 𝑦𝑥)
72, 6mpancom 684 . . 3 (𝐴Ref𝐵 → ∀𝑦𝐴𝑥𝐵 𝑦𝑥)
8 sseq1 3989 . . . . 5 (𝑦 = 𝑆 → (𝑦𝑥𝑆𝑥))
98rexbidv 3294 . . . 4 (𝑦 = 𝑆 → (∃𝑥𝐵 𝑦𝑥 ↔ ∃𝑥𝐵 𝑆𝑥))
109rspccv 3617 . . 3 (∀𝑦𝐴𝑥𝐵 𝑦𝑥 → (𝑆𝐴 → ∃𝑥𝐵 𝑆𝑥))
117, 10syl 17 . 2 (𝐴Ref𝐵 → (𝑆𝐴 → ∃𝑥𝐵 𝑆𝑥))
1211imp 407 1 ((𝐴Ref𝐵𝑆𝐴) → ∃𝑥𝐵 𝑆𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  wrex 3136  Vcvv 3492  wss 3933   cuni 4830   class class class wbr 5057  Refcref 22038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-ref 22041
This theorem is referenced by:  reftr  22050  refun0  22051  refssfne  33603
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