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Theorem heiborlem1 33581
Description: Lemma for heibor 33591. We work with a fixed open cover 𝑈 throughout. The set 𝐾 is the set of all subsets of 𝑋 that admit no finite subcover of 𝑈. (We wish to prove that 𝐾 is empty.) If a set 𝐶 has no finite subcover, then any finite cover of 𝐶 must contain a set that also has no finite subcover. (Contributed by Jeff Madsen, 23-Jan-2014.)
Hypotheses
Ref Expression
heibor.1 𝐽 = (MetOpen‘𝐷)
heibor.3 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣}
heiborlem1.4 𝐵 ∈ V
Assertion
Ref Expression
heiborlem1 ((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵𝐶𝐾) → ∃𝑥𝐴 𝐵𝐾)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑢,𝑣,𝐷   𝑢,𝐵,𝑣   𝑢,𝐽,𝑣,𝑥   𝑢,𝑈,𝑣,𝑥   𝑢,𝐶,𝑣   𝑥,𝐾
Allowed substitution hints:   𝐴(𝑣,𝑢)   𝐵(𝑥)   𝐶(𝑥)   𝐾(𝑣,𝑢)

Proof of Theorem heiborlem1
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 heiborlem1.4 . . . . . . . 8 𝐵 ∈ V
2 sseq1 3618 . . . . . . . . . 10 (𝑢 = 𝐵 → (𝑢 𝑣𝐵 𝑣))
32rexbidv 3048 . . . . . . . . 9 (𝑢 = 𝐵 → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣))
43notbid 308 . . . . . . . 8 (𝑢 = 𝐵 → (¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣))
5 heibor.3 . . . . . . . 8 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣}
61, 4, 5elab2 3348 . . . . . . 7 (𝐵𝐾 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣)
76con2bii 347 . . . . . 6 (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣 ↔ ¬ 𝐵𝐾)
87ralbii 2977 . . . . 5 (∀𝑥𝐴𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣 ↔ ∀𝑥𝐴 ¬ 𝐵𝐾)
9 ralnex 2989 . . . . 5 (∀𝑥𝐴 ¬ 𝐵𝐾 ↔ ¬ ∃𝑥𝐴 𝐵𝐾)
108, 9bitr2i 265 . . . 4 (¬ ∃𝑥𝐴 𝐵𝐾 ↔ ∀𝑥𝐴𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣)
11 unieq 4435 . . . . . . . . 9 (𝑣 = (𝑡𝑥) → 𝑣 = (𝑡𝑥))
1211sseq2d 3625 . . . . . . . 8 (𝑣 = (𝑡𝑥) → (𝐵 𝑣𝐵 (𝑡𝑥)))
1312ac6sfi 8189 . . . . . . 7 ((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣) → ∃𝑡(𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥)))
1413ex 450 . . . . . 6 (𝐴 ∈ Fin → (∀𝑥𝐴𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣 → ∃𝑡(𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))))
1514adantr 481 . . . . 5 ((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) → (∀𝑥𝐴𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣 → ∃𝑡(𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))))
16 sseq1 3618 . . . . . . . . . . . 12 (𝑢 = 𝐶 → (𝑢 𝑣𝐶 𝑣))
1716rexbidv 3048 . . . . . . . . . . 11 (𝑢 = 𝐶 → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 𝑣))
1817notbid 308 . . . . . . . . . 10 (𝑢 = 𝐶 → (¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 𝑣))
1918, 5elab2g 3347 . . . . . . . . 9 (𝐶𝐾 → (𝐶𝐾 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 𝑣))
2019ibi 256 . . . . . . . 8 (𝐶𝐾 → ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 𝑣)
21 frn 6040 . . . . . . . . . . . . . . 15 (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) → ran 𝑡 ⊆ (𝒫 𝑈 ∩ Fin))
2221ad2antrl 763 . . . . . . . . . . . . . 14 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ⊆ (𝒫 𝑈 ∩ Fin))
23 inss1 3825 . . . . . . . . . . . . . 14 (𝒫 𝑈 ∩ Fin) ⊆ 𝒫 𝑈
2422, 23syl6ss 3607 . . . . . . . . . . . . 13 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ⊆ 𝒫 𝑈)
25 sspwuni 4602 . . . . . . . . . . . . 13 (ran 𝑡 ⊆ 𝒫 𝑈 ran 𝑡𝑈)
2624, 25sylib 208 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡𝑈)
27 vex 3198 . . . . . . . . . . . . . . 15 𝑡 ∈ V
2827rnex 7085 . . . . . . . . . . . . . 14 ran 𝑡 ∈ V
2928uniex 6938 . . . . . . . . . . . . 13 ran 𝑡 ∈ V
3029elpw 4155 . . . . . . . . . . . 12 ( ran 𝑡 ∈ 𝒫 𝑈 ran 𝑡𝑈)
3126, 30sylibr 224 . . . . . . . . . . 11 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ∈ 𝒫 𝑈)
32 ffn 6032 . . . . . . . . . . . . . . 15 (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) → 𝑡 Fn 𝐴)
3332ad2antrl 763 . . . . . . . . . . . . . 14 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → 𝑡 Fn 𝐴)
34 dffn4 6108 . . . . . . . . . . . . . 14 (𝑡 Fn 𝐴𝑡:𝐴onto→ran 𝑡)
3533, 34sylib 208 . . . . . . . . . . . . 13 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → 𝑡:𝐴onto→ran 𝑡)
36 fofi 8237 . . . . . . . . . . . . 13 ((𝐴 ∈ Fin ∧ 𝑡:𝐴onto→ran 𝑡) → ran 𝑡 ∈ Fin)
3735, 36syldan 487 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ∈ Fin)
38 inss2 3826 . . . . . . . . . . . . 13 (𝒫 𝑈 ∩ Fin) ⊆ Fin
3922, 38syl6ss 3607 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ⊆ Fin)
40 unifi 8240 . . . . . . . . . . . 12 ((ran 𝑡 ∈ Fin ∧ ran 𝑡 ⊆ Fin) → ran 𝑡 ∈ Fin)
4137, 39, 40syl2anc 692 . . . . . . . . . . 11 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ∈ Fin)
4231, 41elind 3790 . . . . . . . . . 10 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ∈ (𝒫 𝑈 ∩ Fin))
4342adantlr 750 . . . . . . . . 9 (((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ∈ (𝒫 𝑈 ∩ Fin))
44 simplr 791 . . . . . . . . . 10 (((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → 𝐶 𝑥𝐴 𝐵)
45 fnfvelrn 6342 . . . . . . . . . . . . . . . . . 18 ((𝑡 Fn 𝐴𝑥𝐴) → (𝑡𝑥) ∈ ran 𝑡)
4632, 45sylan 488 . . . . . . . . . . . . . . . . 17 ((𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ 𝑥𝐴) → (𝑡𝑥) ∈ ran 𝑡)
4746adantll 749 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ Fin ∧ 𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin)) ∧ 𝑥𝐴) → (𝑡𝑥) ∈ ran 𝑡)
48 elssuni 4458 . . . . . . . . . . . . . . . 16 ((𝑡𝑥) ∈ ran 𝑡 → (𝑡𝑥) ⊆ ran 𝑡)
49 uniss 4449 . . . . . . . . . . . . . . . 16 ((𝑡𝑥) ⊆ ran 𝑡 (𝑡𝑥) ⊆ ran 𝑡)
5047, 48, 493syl 18 . . . . . . . . . . . . . . 15 (((𝐴 ∈ Fin ∧ 𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin)) ∧ 𝑥𝐴) → (𝑡𝑥) ⊆ ran 𝑡)
51 sstr2 3602 . . . . . . . . . . . . . . 15 (𝐵 (𝑡𝑥) → ( (𝑡𝑥) ⊆ ran 𝑡𝐵 ran 𝑡))
5250, 51syl5com 31 . . . . . . . . . . . . . 14 (((𝐴 ∈ Fin ∧ 𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin)) ∧ 𝑥𝐴) → (𝐵 (𝑡𝑥) → 𝐵 ran 𝑡))
5352ralimdva 2959 . . . . . . . . . . . . 13 ((𝐴 ∈ Fin ∧ 𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin)) → (∀𝑥𝐴 𝐵 (𝑡𝑥) → ∀𝑥𝐴 𝐵 ran 𝑡))
5453impr 648 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ∀𝑥𝐴 𝐵 ran 𝑡)
55 iunss 4552 . . . . . . . . . . . 12 ( 𝑥𝐴 𝐵 ran 𝑡 ↔ ∀𝑥𝐴 𝐵 ran 𝑡)
5654, 55sylibr 224 . . . . . . . . . . 11 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → 𝑥𝐴 𝐵 ran 𝑡)
5756adantlr 750 . . . . . . . . . 10 (((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → 𝑥𝐴 𝐵 ran 𝑡)
5844, 57sstrd 3605 . . . . . . . . 9 (((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → 𝐶 ran 𝑡)
59 unieq 4435 . . . . . . . . . . 11 (𝑣 = ran 𝑡 𝑣 = ran 𝑡)
6059sseq2d 3625 . . . . . . . . . 10 (𝑣 = ran 𝑡 → (𝐶 𝑣𝐶 ran 𝑡))
6160rspcev 3304 . . . . . . . . 9 (( ran 𝑡 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝐶 ran 𝑡) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 𝑣)
6243, 58, 61syl2anc 692 . . . . . . . 8 (((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 𝑣)
6320, 62nsyl3 133 . . . . . . 7 (((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ¬ 𝐶𝐾)
6463ex 450 . . . . . 6 ((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) → ((𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥)) → ¬ 𝐶𝐾))
6564exlimdv 1859 . . . . 5 ((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) → (∃𝑡(𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥)) → ¬ 𝐶𝐾))
6615, 65syld 47 . . . 4 ((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) → (∀𝑥𝐴𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣 → ¬ 𝐶𝐾))
6710, 66syl5bi 232 . . 3 ((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) → (¬ ∃𝑥𝐴 𝐵𝐾 → ¬ 𝐶𝐾))
6867con4d 114 . 2 ((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) → (𝐶𝐾 → ∃𝑥𝐴 𝐵𝐾))
69683impia 1259 1 ((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵𝐶𝐾) → ∃𝑥𝐴 𝐵𝐾)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1036   = wceq 1481  wex 1702  wcel 1988  {cab 2606  wral 2909  wrex 2910  Vcvv 3195  cin 3566  wss 3567  𝒫 cpw 4149   cuni 4427   ciun 4511  ran crn 5105   Fn wfn 5871  wf 5872  ontowfo 5874  cfv 5876  Fincfn 7940  MetOpencmopn 19717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-en 7941  df-dom 7942  df-fin 7944
This theorem is referenced by:  heiborlem3  33583  heiborlem10  33590
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