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Theorem refssex 22878
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 22875 . . . . 5 Rel Ref
21brrelex1i 5693 . . . 4 (𝐴Ref𝐵𝐴 ∈ V)
3 eqid 2737 . . . . . 6 𝐴 = 𝐴
4 eqid 2737 . . . . . 6 𝐵 = 𝐵
53, 4isref 22876 . . . . 5 (𝐴 ∈ V → (𝐴Ref𝐵 ↔ ( 𝐵 = 𝐴 ∧ ∀𝑦𝐴𝑥𝐵 𝑦𝑥)))
65simplbda 501 . . . 4 ((𝐴 ∈ V ∧ 𝐴Ref𝐵) → ∀𝑦𝐴𝑥𝐵 𝑦𝑥)
72, 6mpancom 687 . . 3 (𝐴Ref𝐵 → ∀𝑦𝐴𝑥𝐵 𝑦𝑥)
8 sseq1 3974 . . . . 5 (𝑦 = 𝑆 → (𝑦𝑥𝑆𝑥))
98rexbidv 3176 . . . 4 (𝑦 = 𝑆 → (∃𝑥𝐵 𝑦𝑥 ↔ ∃𝑥𝐵 𝑆𝑥))
109rspccv 3581 . . 3 (∀𝑦𝐴𝑥𝐵 𝑦𝑥 → (𝑆𝐴 → ∃𝑥𝐵 𝑆𝑥))
117, 10syl 17 . 2 (𝐴Ref𝐵 → (𝑆𝐴 → ∃𝑥𝐵 𝑆𝑥))
1211imp 408 1 ((𝐴Ref𝐵𝑆𝐴) → ∃𝑥𝐵 𝑆𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3065  wrex 3074  Vcvv 3448  wss 3915   cuni 4870   class class class wbr 5110  Refcref 22869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-xp 5644  df-rel 5645  df-ref 22872
This theorem is referenced by:  reftr  22881  refun0  22882  refssfne  34859
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