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Theorem refssfne 32478
 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 21359 . . . . . . 7 Rel Ref
21brrelex2i 5193 . . . . . 6 (𝐵Ref𝐴𝐴 ∈ V)
32adantl 481 . . . . 5 ((𝑋 = 𝑌𝐵Ref𝐴) → 𝐴 ∈ V)
41brrelexi 5192 . . . . . 6 (𝐵Ref𝐴𝐵 ∈ V)
54adantl 481 . . . . 5 ((𝑋 = 𝑌𝐵Ref𝐴) → 𝐵 ∈ V)
6 unexg 7001 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
73, 5, 6syl2anc 694 . . . 4 ((𝑋 = 𝑌𝐵Ref𝐴) → (𝐴𝐵) ∈ V)
8 ssun2 3810 . . . . . 6 𝐵 ⊆ (𝐴𝐵)
98a1i 11 . . . . 5 ((𝑋 = 𝑌𝐵Ref𝐴) → 𝐵 ⊆ (𝐴𝐵))
10 ssun1 3809 . . . . . . 7 𝐴 ⊆ (𝐴𝐵)
1110a1i 11 . . . . . 6 ((𝑋 = 𝑌𝐵Ref𝐴) → 𝐴 ⊆ (𝐴𝐵))
12 eqimss2 3691 . . . . . . . . 9 (𝑋 = 𝑌𝑌𝑋)
1312adantr 480 . . . . . . . 8 ((𝑋 = 𝑌𝐵Ref𝐴) → 𝑌𝑋)
14 ssequn2 3819 . . . . . . . 8 (𝑌𝑋 ↔ (𝑋𝑌) = 𝑋)
1513, 14sylib 208 . . . . . . 7 ((𝑋 = 𝑌𝐵Ref𝐴) → (𝑋𝑌) = 𝑋)
1615eqcomd 2657 . . . . . 6 ((𝑋 = 𝑌𝐵Ref𝐴) → 𝑋 = (𝑋𝑌))
17 refssfne.1 . . . . . . 7 𝑋 = 𝐴
18 refssfne.2 . . . . . . . . 9 𝑌 = 𝐵
1917, 18uneq12i 3798 . . . . . . . 8 (𝑋𝑌) = ( 𝐴 𝐵)
20 uniun 4488 . . . . . . . 8 (𝐴𝐵) = ( 𝐴 𝐵)
2119, 20eqtr4i 2676 . . . . . . 7 (𝑋𝑌) = (𝐴𝐵)
2217, 21fness 32469 . . . . . 6 (((𝐴𝐵) ∈ V ∧ 𝐴 ⊆ (𝐴𝐵) ∧ 𝑋 = (𝑋𝑌)) → 𝐴Fne(𝐴𝐵))
237, 11, 16, 22syl3anc 1366 . . . . 5 ((𝑋 = 𝑌𝐵Ref𝐴) → 𝐴Fne(𝐴𝐵))
24 elun 3786 . . . . . . . 8 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
25 ssid 3657 . . . . . . . . . . 11 𝑥𝑥
26 sseq2 3660 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝑥𝑦𝑥𝑥))
2726rspcev 3340 . . . . . . . . . . 11 ((𝑥𝐴𝑥𝑥) → ∃𝑦𝐴 𝑥𝑦)
2825, 27mpan2 707 . . . . . . . . . 10 (𝑥𝐴 → ∃𝑦𝐴 𝑥𝑦)
2928a1i 11 . . . . . . . . 9 ((𝑋 = 𝑌𝐵Ref𝐴) → (𝑥𝐴 → ∃𝑦𝐴 𝑥𝑦))
30 refssex 21362 . . . . . . . . . . 11 ((𝐵Ref𝐴𝑥𝐵) → ∃𝑦𝐴 𝑥𝑦)
3130ex 449 . . . . . . . . . 10 (𝐵Ref𝐴 → (𝑥𝐵 → ∃𝑦𝐴 𝑥𝑦))
3231adantl 481 . . . . . . . . 9 ((𝑋 = 𝑌𝐵Ref𝐴) → (𝑥𝐵 → ∃𝑦𝐴 𝑥𝑦))
3329, 32jaod 394 . . . . . . . 8 ((𝑋 = 𝑌𝐵Ref𝐴) → ((𝑥𝐴𝑥𝐵) → ∃𝑦𝐴 𝑥𝑦))
3424, 33syl5bi 232 . . . . . . 7 ((𝑋 = 𝑌𝐵Ref𝐴) → (𝑥 ∈ (𝐴𝐵) → ∃𝑦𝐴 𝑥𝑦))
3534ralrimiv 2994 . . . . . 6 ((𝑋 = 𝑌𝐵Ref𝐴) → ∀𝑥 ∈ (𝐴𝐵)∃𝑦𝐴 𝑥𝑦)
3621, 17isref 21360 . . . . . . 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 977 . . . . 5 ((𝑋 = 𝑌𝐵Ref𝐴) → (𝐴𝐵)Ref𝐴)
399, 23, 38jca32 557 . . . 4 ((𝑋 = 𝑌𝐵Ref𝐴) → (𝐵 ⊆ (𝐴𝐵) ∧ (𝐴Fne(𝐴𝐵) ∧ (𝐴𝐵)Ref𝐴)))
40 sseq2 3660 . . . . . 6 (𝑐 = (𝐴𝐵) → (𝐵𝑐𝐵 ⊆ (𝐴𝐵)))
41 breq2 4689 . . . . . . 7 (𝑐 = (𝐴𝐵) → (𝐴Fne𝑐𝐴Fne(𝐴𝐵)))
42 breq1 4688 . . . . . . 7 (𝑐 = (𝐴𝐵) → (𝑐Ref𝐴 ↔ (𝐴𝐵)Ref𝐴))
4341, 42anbi12d 747 . . . . . 6 (𝑐 = (𝐴𝐵) → ((𝐴Fne𝑐𝑐Ref𝐴) ↔ (𝐴Fne(𝐴𝐵) ∧ (𝐴𝐵)Ref𝐴)))
4440, 43anbi12d 747 . . . . 5 (𝑐 = (𝐴𝐵) → ((𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴)) ↔ (𝐵 ⊆ (𝐴𝐵) ∧ (𝐴Fne(𝐴𝐵) ∧ (𝐴𝐵)Ref𝐴))))
4544spcegv 3325 . . . 4 ((𝐴𝐵) ∈ V → ((𝐵 ⊆ (𝐴𝐵) ∧ (𝐴Fne(𝐴𝐵) ∧ (𝐴𝐵)Ref𝐴)) → ∃𝑐(𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))))
467, 39, 45sylc 65 . . 3 ((𝑋 = 𝑌𝐵Ref𝐴) → ∃𝑐(𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴)))
4746ex 449 . 2 (𝑋 = 𝑌 → (𝐵Ref𝐴 → ∃𝑐(𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))))
48 vex 3234 . . . . . . . 8 𝑐 ∈ V
4948ssex 4835 . . . . . . 7 (𝐵𝑐𝐵 ∈ V)
5049ad2antrl 764 . . . . . 6 ((𝑋 = 𝑌 ∧ (𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝐵 ∈ V)
51 simprl 809 . . . . . 6 ((𝑋 = 𝑌 ∧ (𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝐵𝑐)
52 simpl 472 . . . . . . 7 ((𝑋 = 𝑌 ∧ (𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝑋 = 𝑌)
53 eqid 2651 . . . . . . . . . 10 𝑐 = 𝑐
5453, 17refbas 21361 . . . . . . . . 9 (𝑐Ref𝐴𝑋 = 𝑐)
5554adantl 481 . . . . . . . 8 ((𝐴Fne𝑐𝑐Ref𝐴) → 𝑋 = 𝑐)
5655ad2antll 765 . . . . . . 7 ((𝑋 = 𝑌 ∧ (𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝑋 = 𝑐)
5752, 56eqtr3d 2687 . . . . . 6 ((𝑋 = 𝑌 ∧ (𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝑌 = 𝑐)
5818, 53ssref 21363 . . . . . 6 ((𝐵 ∈ V ∧ 𝐵𝑐𝑌 = 𝑐) → 𝐵Ref𝑐)
5950, 51, 57, 58syl3anc 1366 . . . . 5 ((𝑋 = 𝑌 ∧ (𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝐵Ref𝑐)
60 simprrr 822 . . . . 5 ((𝑋 = 𝑌 ∧ (𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝑐Ref𝐴)
61 reftr 21365 . . . . 5 ((𝐵Ref𝑐𝑐Ref𝐴) → 𝐵Ref𝐴)
6259, 60, 61syl2anc 694 . . . 4 ((𝑋 = 𝑌 ∧ (𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝐵Ref𝐴)
6362ex 449 . . 3 (𝑋 = 𝑌 → ((𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴)) → 𝐵Ref𝐴))
6463exlimdv 1901 . 2 (𝑋 = 𝑌 → (∃𝑐(𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴)) → 𝐵Ref𝐴))
6547, 64impbid 202 1 (𝑋 = 𝑌 → (𝐵Ref𝐴 ↔ ∃𝑐(𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 382   ∧ wa 383   = wceq 1523  ∃wex 1744   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942  Vcvv 3231   ∪ cun 3605   ⊆ wss 3607  ∪ cuni 4468   class class class wbr 4685  Refcref 21353  Fnecfne 32456 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-topgen 16151  df-ref 21356  df-fne 32457 This theorem is referenced by: (None)
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