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Theorem refssfne 35547
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 23233 . . . . . . 7 Rel Ref
21brrelex2i 5734 . . . . . 6 (𝐵Ref𝐴𝐴 ∈ V)
32adantl 481 . . . . 5 ((𝑋 = 𝑌𝐵Ref𝐴) → 𝐴 ∈ V)
41brrelex1i 5733 . . . . . 6 (𝐵Ref𝐴𝐵 ∈ V)
54adantl 481 . . . . 5 ((𝑋 = 𝑌𝐵Ref𝐴) → 𝐵 ∈ V)
6 unexg 7739 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
73, 5, 6syl2anc 583 . . . 4 ((𝑋 = 𝑌𝐵Ref𝐴) → (𝐴𝐵) ∈ V)
8 ssun2 4174 . . . . . 6 𝐵 ⊆ (𝐴𝐵)
98a1i 11 . . . . 5 ((𝑋 = 𝑌𝐵Ref𝐴) → 𝐵 ⊆ (𝐴𝐵))
10 ssun1 4173 . . . . . . 7 𝐴 ⊆ (𝐴𝐵)
1110a1i 11 . . . . . 6 ((𝑋 = 𝑌𝐵Ref𝐴) → 𝐴 ⊆ (𝐴𝐵))
12 eqimss2 4042 . . . . . . . . 9 (𝑋 = 𝑌𝑌𝑋)
1312adantr 480 . . . . . . . 8 ((𝑋 = 𝑌𝐵Ref𝐴) → 𝑌𝑋)
14 ssequn2 4184 . . . . . . . 8 (𝑌𝑋 ↔ (𝑋𝑌) = 𝑋)
1513, 14sylib 217 . . . . . . 7 ((𝑋 = 𝑌𝐵Ref𝐴) → (𝑋𝑌) = 𝑋)
1615eqcomd 2737 . . . . . 6 ((𝑋 = 𝑌𝐵Ref𝐴) → 𝑋 = (𝑋𝑌))
17 refssfne.1 . . . . . . 7 𝑋 = 𝐴
18 refssfne.2 . . . . . . . . 9 𝑌 = 𝐵
1917, 18uneq12i 4162 . . . . . . . 8 (𝑋𝑌) = ( 𝐴 𝐵)
20 uniun 4935 . . . . . . . 8 (𝐴𝐵) = ( 𝐴 𝐵)
2119, 20eqtr4i 2762 . . . . . . 7 (𝑋𝑌) = (𝐴𝐵)
2217, 21fness 35538 . . . . . 6 (((𝐴𝐵) ∈ V ∧ 𝐴 ⊆ (𝐴𝐵) ∧ 𝑋 = (𝑋𝑌)) → 𝐴Fne(𝐴𝐵))
237, 11, 16, 22syl3anc 1370 . . . . 5 ((𝑋 = 𝑌𝐵Ref𝐴) → 𝐴Fne(𝐴𝐵))
24 elun 4149 . . . . . . . 8 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
25 ssid 4005 . . . . . . . . . . 11 𝑥𝑥
26 sseq2 4009 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝑥𝑦𝑥𝑥))
2726rspcev 3613 . . . . . . . . . . 11 ((𝑥𝐴𝑥𝑥) → ∃𝑦𝐴 𝑥𝑦)
2825, 27mpan2 688 . . . . . . . . . 10 (𝑥𝐴 → ∃𝑦𝐴 𝑥𝑦)
2928a1i 11 . . . . . . . . 9 ((𝑋 = 𝑌𝐵Ref𝐴) → (𝑥𝐴 → ∃𝑦𝐴 𝑥𝑦))
30 refssex 23236 . . . . . . . . . . 11 ((𝐵Ref𝐴𝑥𝐵) → ∃𝑦𝐴 𝑥𝑦)
3130ex 412 . . . . . . . . . 10 (𝐵Ref𝐴 → (𝑥𝐵 → ∃𝑦𝐴 𝑥𝑦))
3231adantl 481 . . . . . . . . 9 ((𝑋 = 𝑌𝐵Ref𝐴) → (𝑥𝐵 → ∃𝑦𝐴 𝑥𝑦))
3329, 32jaod 856 . . . . . . . 8 ((𝑋 = 𝑌𝐵Ref𝐴) → ((𝑥𝐴𝑥𝐵) → ∃𝑦𝐴 𝑥𝑦))
3424, 33biimtrid 241 . . . . . . 7 ((𝑋 = 𝑌𝐵Ref𝐴) → (𝑥 ∈ (𝐴𝐵) → ∃𝑦𝐴 𝑥𝑦))
3534ralrimiv 3144 . . . . . 6 ((𝑋 = 𝑌𝐵Ref𝐴) → ∀𝑥 ∈ (𝐴𝐵)∃𝑦𝐴 𝑥𝑦)
3621, 17isref 23234 . . . . . . 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 710 . . . . 5 ((𝑋 = 𝑌𝐵Ref𝐴) → (𝐴𝐵)Ref𝐴)
399, 23, 38jca32 515 . . . 4 ((𝑋 = 𝑌𝐵Ref𝐴) → (𝐵 ⊆ (𝐴𝐵) ∧ (𝐴Fne(𝐴𝐵) ∧ (𝐴𝐵)Ref𝐴)))
40 sseq2 4009 . . . . . 6 (𝑐 = (𝐴𝐵) → (𝐵𝑐𝐵 ⊆ (𝐴𝐵)))
41 breq2 5153 . . . . . . 7 (𝑐 = (𝐴𝐵) → (𝐴Fne𝑐𝐴Fne(𝐴𝐵)))
42 breq1 5152 . . . . . . 7 (𝑐 = (𝐴𝐵) → (𝑐Ref𝐴 ↔ (𝐴𝐵)Ref𝐴))
4341, 42anbi12d 630 . . . . . 6 (𝑐 = (𝐴𝐵) → ((𝐴Fne𝑐𝑐Ref𝐴) ↔ (𝐴Fne(𝐴𝐵) ∧ (𝐴𝐵)Ref𝐴)))
4440, 43anbi12d 630 . . . . 5 (𝑐 = (𝐴𝐵) → ((𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴)) ↔ (𝐵 ⊆ (𝐴𝐵) ∧ (𝐴Fne(𝐴𝐵) ∧ (𝐴𝐵)Ref𝐴))))
4544spcegv 3588 . . . 4 ((𝐴𝐵) ∈ V → ((𝐵 ⊆ (𝐴𝐵) ∧ (𝐴Fne(𝐴𝐵) ∧ (𝐴𝐵)Ref𝐴)) → ∃𝑐(𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))))
467, 39, 45sylc 65 . . 3 ((𝑋 = 𝑌𝐵Ref𝐴) → ∃𝑐(𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴)))
4746ex 412 . 2 (𝑋 = 𝑌 → (𝐵Ref𝐴 → ∃𝑐(𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))))
48 vex 3477 . . . . . . . 8 𝑐 ∈ V
4948ssex 5322 . . . . . . 7 (𝐵𝑐𝐵 ∈ V)
5049ad2antrl 725 . . . . . 6 ((𝑋 = 𝑌 ∧ (𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝐵 ∈ V)
51 simprl 768 . . . . . 6 ((𝑋 = 𝑌 ∧ (𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝐵𝑐)
52 simpl 482 . . . . . . 7 ((𝑋 = 𝑌 ∧ (𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝑋 = 𝑌)
53 eqid 2731 . . . . . . . . . 10 𝑐 = 𝑐
5453, 17refbas 23235 . . . . . . . . 9 (𝑐Ref𝐴𝑋 = 𝑐)
5554adantl 481 . . . . . . . 8 ((𝐴Fne𝑐𝑐Ref𝐴) → 𝑋 = 𝑐)
5655ad2antll 726 . . . . . . 7 ((𝑋 = 𝑌 ∧ (𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝑋 = 𝑐)
5752, 56eqtr3d 2773 . . . . . 6 ((𝑋 = 𝑌 ∧ (𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝑌 = 𝑐)
5818, 53ssref 23237 . . . . . 6 ((𝐵 ∈ V ∧ 𝐵𝑐𝑌 = 𝑐) → 𝐵Ref𝑐)
5950, 51, 57, 58syl3anc 1370 . . . . 5 ((𝑋 = 𝑌 ∧ (𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝐵Ref𝑐)
60 simprrr 779 . . . . 5 ((𝑋 = 𝑌 ∧ (𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝑐Ref𝐴)
61 reftr 23239 . . . . 5 ((𝐵Ref𝑐𝑐Ref𝐴) → 𝐵Ref𝐴)
6259, 60, 61syl2anc 583 . . . 4 ((𝑋 = 𝑌 ∧ (𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝐵Ref𝐴)
6362ex 412 . . 3 (𝑋 = 𝑌 → ((𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴)) → 𝐵Ref𝐴))
6463exlimdv 1935 . 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 395  wo 844   = wceq 1540  wex 1780  wcel 2105  wral 3060  wrex 3069  Vcvv 3473  cun 3947  wss 3949   cuni 4909   class class class wbr 5149  Refcref 23227  Fnecfne 35525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-topgen 17394  df-ref 23230  df-fne 35526
This theorem is referenced by: (None)
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