Users' Mathboxes Mathbox for Jeff Hankins < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  refssfne Structured version   Visualization version   GIF version

Theorem refssfne 35546
Description: A cover is a refinement iff it is a subcover of something which is both finer and a refinement. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)
Hypotheses
Ref Expression
refssfne.1 𝑋 = 𝐴
refssfne.2 𝑌 = 𝐵
Assertion
Ref Expression
refssfne (𝑋 = 𝑌 → (𝐵Ref𝐴 ↔ ∃𝑐(𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))))
Distinct variable groups:   𝐴,𝑐   𝐵,𝑐   𝑋,𝑐   𝑌,𝑐

Proof of Theorem refssfne
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 refrel 23232 . . . . . . 7 Rel Ref
21brrelex2i 5732 . . . . . 6 (𝐵Ref𝐴𝐴 ∈ V)
32adantl 480 . . . . 5 ((𝑋 = 𝑌𝐵Ref𝐴) → 𝐴 ∈ V)
41brrelex1i 5731 . . . . . 6 (𝐵Ref𝐴𝐵 ∈ V)
54adantl 480 . . . . 5 ((𝑋 = 𝑌𝐵Ref𝐴) → 𝐵 ∈ V)
6 unexg 7738 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
73, 5, 6syl2anc 582 . . . 4 ((𝑋 = 𝑌𝐵Ref𝐴) → (𝐴𝐵) ∈ V)
8 ssun2 4172 . . . . . 6 𝐵 ⊆ (𝐴𝐵)
98a1i 11 . . . . 5 ((𝑋 = 𝑌𝐵Ref𝐴) → 𝐵 ⊆ (𝐴𝐵))
10 ssun1 4171 . . . . . . 7 𝐴 ⊆ (𝐴𝐵)
1110a1i 11 . . . . . 6 ((𝑋 = 𝑌𝐵Ref𝐴) → 𝐴 ⊆ (𝐴𝐵))
12 eqimss2 4040 . . . . . . . . 9 (𝑋 = 𝑌𝑌𝑋)
1312adantr 479 . . . . . . . 8 ((𝑋 = 𝑌𝐵Ref𝐴) → 𝑌𝑋)
14 ssequn2 4182 . . . . . . . 8 (𝑌𝑋 ↔ (𝑋𝑌) = 𝑋)
1513, 14sylib 217 . . . . . . 7 ((𝑋 = 𝑌𝐵Ref𝐴) → (𝑋𝑌) = 𝑋)
1615eqcomd 2736 . . . . . 6 ((𝑋 = 𝑌𝐵Ref𝐴) → 𝑋 = (𝑋𝑌))
17 refssfne.1 . . . . . . 7 𝑋 = 𝐴
18 refssfne.2 . . . . . . . . 9 𝑌 = 𝐵
1917, 18uneq12i 4160 . . . . . . . 8 (𝑋𝑌) = ( 𝐴 𝐵)
20 uniun 4933 . . . . . . . 8 (𝐴𝐵) = ( 𝐴 𝐵)
2119, 20eqtr4i 2761 . . . . . . 7 (𝑋𝑌) = (𝐴𝐵)
2217, 21fness 35537 . . . . . 6 (((𝐴𝐵) ∈ V ∧ 𝐴 ⊆ (𝐴𝐵) ∧ 𝑋 = (𝑋𝑌)) → 𝐴Fne(𝐴𝐵))
237, 11, 16, 22syl3anc 1369 . . . . 5 ((𝑋 = 𝑌𝐵Ref𝐴) → 𝐴Fne(𝐴𝐵))
24 elun 4147 . . . . . . . 8 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
25 ssid 4003 . . . . . . . . . . 11 𝑥𝑥
26 sseq2 4007 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝑥𝑦𝑥𝑥))
2726rspcev 3611 . . . . . . . . . . 11 ((𝑥𝐴𝑥𝑥) → ∃𝑦𝐴 𝑥𝑦)
2825, 27mpan2 687 . . . . . . . . . 10 (𝑥𝐴 → ∃𝑦𝐴 𝑥𝑦)
2928a1i 11 . . . . . . . . 9 ((𝑋 = 𝑌𝐵Ref𝐴) → (𝑥𝐴 → ∃𝑦𝐴 𝑥𝑦))
30 refssex 23235 . . . . . . . . . . 11 ((𝐵Ref𝐴𝑥𝐵) → ∃𝑦𝐴 𝑥𝑦)
3130ex 411 . . . . . . . . . 10 (𝐵Ref𝐴 → (𝑥𝐵 → ∃𝑦𝐴 𝑥𝑦))
3231adantl 480 . . . . . . . . 9 ((𝑋 = 𝑌𝐵Ref𝐴) → (𝑥𝐵 → ∃𝑦𝐴 𝑥𝑦))
3329, 32jaod 855 . . . . . . . 8 ((𝑋 = 𝑌𝐵Ref𝐴) → ((𝑥𝐴𝑥𝐵) → ∃𝑦𝐴 𝑥𝑦))
3424, 33biimtrid 241 . . . . . . 7 ((𝑋 = 𝑌𝐵Ref𝐴) → (𝑥 ∈ (𝐴𝐵) → ∃𝑦𝐴 𝑥𝑦))
3534ralrimiv 3143 . . . . . 6 ((𝑋 = 𝑌𝐵Ref𝐴) → ∀𝑥 ∈ (𝐴𝐵)∃𝑦𝐴 𝑥𝑦)
3621, 17isref 23233 . . . . . . 7 ((𝐴𝐵) ∈ V → ((𝐴𝐵)Ref𝐴 ↔ (𝑋 = (𝑋𝑌) ∧ ∀𝑥 ∈ (𝐴𝐵)∃𝑦𝐴 𝑥𝑦)))
377, 36syl 17 . . . . . 6 ((𝑋 = 𝑌𝐵Ref𝐴) → ((𝐴𝐵)Ref𝐴 ↔ (𝑋 = (𝑋𝑌) ∧ ∀𝑥 ∈ (𝐴𝐵)∃𝑦𝐴 𝑥𝑦)))
3816, 35, 37mpbir2and 709 . . . . 5 ((𝑋 = 𝑌𝐵Ref𝐴) → (𝐴𝐵)Ref𝐴)
399, 23, 38jca32 514 . . . 4 ((𝑋 = 𝑌𝐵Ref𝐴) → (𝐵 ⊆ (𝐴𝐵) ∧ (𝐴Fne(𝐴𝐵) ∧ (𝐴𝐵)Ref𝐴)))
40 sseq2 4007 . . . . . 6 (𝑐 = (𝐴𝐵) → (𝐵𝑐𝐵 ⊆ (𝐴𝐵)))
41 breq2 5151 . . . . . . 7 (𝑐 = (𝐴𝐵) → (𝐴Fne𝑐𝐴Fne(𝐴𝐵)))
42 breq1 5150 . . . . . . 7 (𝑐 = (𝐴𝐵) → (𝑐Ref𝐴 ↔ (𝐴𝐵)Ref𝐴))
4341, 42anbi12d 629 . . . . . 6 (𝑐 = (𝐴𝐵) → ((𝐴Fne𝑐𝑐Ref𝐴) ↔ (𝐴Fne(𝐴𝐵) ∧ (𝐴𝐵)Ref𝐴)))
4440, 43anbi12d 629 . . . . 5 (𝑐 = (𝐴𝐵) → ((𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴)) ↔ (𝐵 ⊆ (𝐴𝐵) ∧ (𝐴Fne(𝐴𝐵) ∧ (𝐴𝐵)Ref𝐴))))
4544spcegv 3586 . . . 4 ((𝐴𝐵) ∈ V → ((𝐵 ⊆ (𝐴𝐵) ∧ (𝐴Fne(𝐴𝐵) ∧ (𝐴𝐵)Ref𝐴)) → ∃𝑐(𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))))
467, 39, 45sylc 65 . . 3 ((𝑋 = 𝑌𝐵Ref𝐴) → ∃𝑐(𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴)))
4746ex 411 . 2 (𝑋 = 𝑌 → (𝐵Ref𝐴 → ∃𝑐(𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))))
48 vex 3476 . . . . . . . 8 𝑐 ∈ V
4948ssex 5320 . . . . . . 7 (𝐵𝑐𝐵 ∈ V)
5049ad2antrl 724 . . . . . 6 ((𝑋 = 𝑌 ∧ (𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝐵 ∈ V)
51 simprl 767 . . . . . 6 ((𝑋 = 𝑌 ∧ (𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝐵𝑐)
52 simpl 481 . . . . . . 7 ((𝑋 = 𝑌 ∧ (𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝑋 = 𝑌)
53 eqid 2730 . . . . . . . . . 10 𝑐 = 𝑐
5453, 17refbas 23234 . . . . . . . . 9 (𝑐Ref𝐴𝑋 = 𝑐)
5554adantl 480 . . . . . . . 8 ((𝐴Fne𝑐𝑐Ref𝐴) → 𝑋 = 𝑐)
5655ad2antll 725 . . . . . . 7 ((𝑋 = 𝑌 ∧ (𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝑋 = 𝑐)
5752, 56eqtr3d 2772 . . . . . 6 ((𝑋 = 𝑌 ∧ (𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝑌 = 𝑐)
5818, 53ssref 23236 . . . . . 6 ((𝐵 ∈ V ∧ 𝐵𝑐𝑌 = 𝑐) → 𝐵Ref𝑐)
5950, 51, 57, 58syl3anc 1369 . . . . 5 ((𝑋 = 𝑌 ∧ (𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝐵Ref𝑐)
60 simprrr 778 . . . . 5 ((𝑋 = 𝑌 ∧ (𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝑐Ref𝐴)
61 reftr 23238 . . . . 5 ((𝐵Ref𝑐𝑐Ref𝐴) → 𝐵Ref𝐴)
6259, 60, 61syl2anc 582 . . . 4 ((𝑋 = 𝑌 ∧ (𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝐵Ref𝐴)
6362ex 411 . . 3 (𝑋 = 𝑌 → ((𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴)) → 𝐵Ref𝐴))
6463exlimdv 1934 . 2 (𝑋 = 𝑌 → (∃𝑐(𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴)) → 𝐵Ref𝐴))
6547, 64impbid 211 1 (𝑋 = 𝑌 → (𝐵Ref𝐴 ↔ ∃𝑐(𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wo 843   = wceq 1539  wex 1779  wcel 2104  wral 3059  wrex 3068  Vcvv 3472  cun 3945  wss 3947   cuni 4907   class class class wbr 5147  Refcref 23226  Fnecfne 35524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-topgen 17393  df-ref 23229  df-fne 35525
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator