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Theorem refssfne 33694
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 22108 . . . . . . 7 Rel Ref
21brrelex2i 5602 . . . . . 6 (𝐵Ref𝐴𝐴 ∈ V)
32adantl 484 . . . . 5 ((𝑋 = 𝑌𝐵Ref𝐴) → 𝐴 ∈ V)
41brrelex1i 5601 . . . . . 6 (𝐵Ref𝐴𝐵 ∈ V)
54adantl 484 . . . . 5 ((𝑋 = 𝑌𝐵Ref𝐴) → 𝐵 ∈ V)
6 unexg 7464 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
73, 5, 6syl2anc 586 . . . 4 ((𝑋 = 𝑌𝐵Ref𝐴) → (𝐴𝐵) ∈ V)
8 ssun2 4147 . . . . . 6 𝐵 ⊆ (𝐴𝐵)
98a1i 11 . . . . 5 ((𝑋 = 𝑌𝐵Ref𝐴) → 𝐵 ⊆ (𝐴𝐵))
10 ssun1 4146 . . . . . . 7 𝐴 ⊆ (𝐴𝐵)
1110a1i 11 . . . . . 6 ((𝑋 = 𝑌𝐵Ref𝐴) → 𝐴 ⊆ (𝐴𝐵))
12 eqimss2 4022 . . . . . . . . 9 (𝑋 = 𝑌𝑌𝑋)
1312adantr 483 . . . . . . . 8 ((𝑋 = 𝑌𝐵Ref𝐴) → 𝑌𝑋)
14 ssequn2 4157 . . . . . . . 8 (𝑌𝑋 ↔ (𝑋𝑌) = 𝑋)
1513, 14sylib 220 . . . . . . 7 ((𝑋 = 𝑌𝐵Ref𝐴) → (𝑋𝑌) = 𝑋)
1615eqcomd 2825 . . . . . 6 ((𝑋 = 𝑌𝐵Ref𝐴) → 𝑋 = (𝑋𝑌))
17 refssfne.1 . . . . . . 7 𝑋 = 𝐴
18 refssfne.2 . . . . . . . . 9 𝑌 = 𝐵
1917, 18uneq12i 4135 . . . . . . . 8 (𝑋𝑌) = ( 𝐴 𝐵)
20 uniun 4849 . . . . . . . 8 (𝐴𝐵) = ( 𝐴 𝐵)
2119, 20eqtr4i 2845 . . . . . . 7 (𝑋𝑌) = (𝐴𝐵)
2217, 21fness 33685 . . . . . 6 (((𝐴𝐵) ∈ V ∧ 𝐴 ⊆ (𝐴𝐵) ∧ 𝑋 = (𝑋𝑌)) → 𝐴Fne(𝐴𝐵))
237, 11, 16, 22syl3anc 1365 . . . . 5 ((𝑋 = 𝑌𝐵Ref𝐴) → 𝐴Fne(𝐴𝐵))
24 elun 4123 . . . . . . . 8 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
25 ssid 3987 . . . . . . . . . . 11 𝑥𝑥
26 sseq2 3991 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝑥𝑦𝑥𝑥))
2726rspcev 3621 . . . . . . . . . . 11 ((𝑥𝐴𝑥𝑥) → ∃𝑦𝐴 𝑥𝑦)
2825, 27mpan2 689 . . . . . . . . . 10 (𝑥𝐴 → ∃𝑦𝐴 𝑥𝑦)
2928a1i 11 . . . . . . . . 9 ((𝑋 = 𝑌𝐵Ref𝐴) → (𝑥𝐴 → ∃𝑦𝐴 𝑥𝑦))
30 refssex 22111 . . . . . . . . . . 11 ((𝐵Ref𝐴𝑥𝐵) → ∃𝑦𝐴 𝑥𝑦)
3130ex 415 . . . . . . . . . 10 (𝐵Ref𝐴 → (𝑥𝐵 → ∃𝑦𝐴 𝑥𝑦))
3231adantl 484 . . . . . . . . 9 ((𝑋 = 𝑌𝐵Ref𝐴) → (𝑥𝐵 → ∃𝑦𝐴 𝑥𝑦))
3329, 32jaod 855 . . . . . . . 8 ((𝑋 = 𝑌𝐵Ref𝐴) → ((𝑥𝐴𝑥𝐵) → ∃𝑦𝐴 𝑥𝑦))
3424, 33syl5bi 244 . . . . . . 7 ((𝑋 = 𝑌𝐵Ref𝐴) → (𝑥 ∈ (𝐴𝐵) → ∃𝑦𝐴 𝑥𝑦))
3534ralrimiv 3179 . . . . . 6 ((𝑋 = 𝑌𝐵Ref𝐴) → ∀𝑥 ∈ (𝐴𝐵)∃𝑦𝐴 𝑥𝑦)
3621, 17isref 22109 . . . . . . 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 711 . . . . 5 ((𝑋 = 𝑌𝐵Ref𝐴) → (𝐴𝐵)Ref𝐴)
399, 23, 38jca32 518 . . . 4 ((𝑋 = 𝑌𝐵Ref𝐴) → (𝐵 ⊆ (𝐴𝐵) ∧ (𝐴Fne(𝐴𝐵) ∧ (𝐴𝐵)Ref𝐴)))
40 sseq2 3991 . . . . . 6 (𝑐 = (𝐴𝐵) → (𝐵𝑐𝐵 ⊆ (𝐴𝐵)))
41 breq2 5061 . . . . . . 7 (𝑐 = (𝐴𝐵) → (𝐴Fne𝑐𝐴Fne(𝐴𝐵)))
42 breq1 5060 . . . . . . 7 (𝑐 = (𝐴𝐵) → (𝑐Ref𝐴 ↔ (𝐴𝐵)Ref𝐴))
4341, 42anbi12d 632 . . . . . 6 (𝑐 = (𝐴𝐵) → ((𝐴Fne𝑐𝑐Ref𝐴) ↔ (𝐴Fne(𝐴𝐵) ∧ (𝐴𝐵)Ref𝐴)))
4440, 43anbi12d 632 . . . . 5 (𝑐 = (𝐴𝐵) → ((𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴)) ↔ (𝐵 ⊆ (𝐴𝐵) ∧ (𝐴Fne(𝐴𝐵) ∧ (𝐴𝐵)Ref𝐴))))
4544spcegv 3595 . . . 4 ((𝐴𝐵) ∈ V → ((𝐵 ⊆ (𝐴𝐵) ∧ (𝐴Fne(𝐴𝐵) ∧ (𝐴𝐵)Ref𝐴)) → ∃𝑐(𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))))
467, 39, 45sylc 65 . . 3 ((𝑋 = 𝑌𝐵Ref𝐴) → ∃𝑐(𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴)))
4746ex 415 . 2 (𝑋 = 𝑌 → (𝐵Ref𝐴 → ∃𝑐(𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))))
48 vex 3496 . . . . . . . 8 𝑐 ∈ V
4948ssex 5216 . . . . . . 7 (𝐵𝑐𝐵 ∈ V)
5049ad2antrl 726 . . . . . 6 ((𝑋 = 𝑌 ∧ (𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝐵 ∈ V)
51 simprl 769 . . . . . 6 ((𝑋 = 𝑌 ∧ (𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝐵𝑐)
52 simpl 485 . . . . . . 7 ((𝑋 = 𝑌 ∧ (𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝑋 = 𝑌)
53 eqid 2819 . . . . . . . . . 10 𝑐 = 𝑐
5453, 17refbas 22110 . . . . . . . . 9 (𝑐Ref𝐴𝑋 = 𝑐)
5554adantl 484 . . . . . . . 8 ((𝐴Fne𝑐𝑐Ref𝐴) → 𝑋 = 𝑐)
5655ad2antll 727 . . . . . . 7 ((𝑋 = 𝑌 ∧ (𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝑋 = 𝑐)
5752, 56eqtr3d 2856 . . . . . 6 ((𝑋 = 𝑌 ∧ (𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝑌 = 𝑐)
5818, 53ssref 22112 . . . . . 6 ((𝐵 ∈ V ∧ 𝐵𝑐𝑌 = 𝑐) → 𝐵Ref𝑐)
5950, 51, 57, 58syl3anc 1365 . . . . 5 ((𝑋 = 𝑌 ∧ (𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝐵Ref𝑐)
60 simprrr 780 . . . . 5 ((𝑋 = 𝑌 ∧ (𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝑐Ref𝐴)
61 reftr 22114 . . . . 5 ((𝐵Ref𝑐𝑐Ref𝐴) → 𝐵Ref𝐴)
6259, 60, 61syl2anc 586 . . . 4 ((𝑋 = 𝑌 ∧ (𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝐵Ref𝐴)
6362ex 415 . . 3 (𝑋 = 𝑌 → ((𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴)) → 𝐵Ref𝐴))
6463exlimdv 1927 . 2 (𝑋 = 𝑌 → (∃𝑐(𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴)) → 𝐵Ref𝐴))
6547, 64impbid 214 1 (𝑋 = 𝑌 → (𝐵Ref𝐴 ↔ ∃𝑐(𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wo 843   = wceq 1530  wex 1773  wcel 2107  wral 3136  wrex 3137  Vcvv 3493  cun 3932  wss 3934   cuni 4830   class class class wbr 5057  Refcref 22102  Fnecfne 33672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-topgen 16709  df-ref 22105  df-fne 33673
This theorem is referenced by: (None)
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