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Theorem heiborlem1 37798
Description: Lemma for heibor 37808. 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 4021 . . . . . . . . . 10 (𝑢 = 𝐵 → (𝑢 𝑣𝐵 𝑣))
32rexbidv 3177 . . . . . . . . 9 (𝑢 = 𝐵 → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣))
43notbid 318 . . . . . . . 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 3685 . . . . . . 7 (𝐵𝐾 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣)
76con2bii 357 . . . . . 6 (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣 ↔ ¬ 𝐵𝐾)
87ralbii 3091 . . . . 5 (∀𝑥𝐴𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣 ↔ ∀𝑥𝐴 ¬ 𝐵𝐾)
9 ralnex 3070 . . . . 5 (∀𝑥𝐴 ¬ 𝐵𝐾 ↔ ¬ ∃𝑥𝐴 𝐵𝐾)
108, 9bitr2i 276 . . . 4 (¬ ∃𝑥𝐴 𝐵𝐾 ↔ ∀𝑥𝐴𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣)
11 unieq 4923 . . . . . . . . 9 (𝑣 = (𝑡𝑥) → 𝑣 = (𝑡𝑥))
1211sseq2d 4028 . . . . . . . 8 (𝑣 = (𝑡𝑥) → (𝐵 𝑣𝐵 (𝑡𝑥)))
1312ac6sfi 9318 . . . . . . 7 ((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣) → ∃𝑡(𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥)))
1413ex 412 . . . . . 6 (𝐴 ∈ Fin → (∀𝑥𝐴𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣 → ∃𝑡(𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))))
1514adantr 480 . . . . 5 ((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) → (∀𝑥𝐴𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣 → ∃𝑡(𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))))
16 sseq1 4021 . . . . . . . . . . . 12 (𝑢 = 𝐶 → (𝑢 𝑣𝐶 𝑣))
1716rexbidv 3177 . . . . . . . . . . 11 (𝑢 = 𝐶 → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 𝑣))
1817notbid 318 . . . . . . . . . 10 (𝑢 = 𝐶 → (¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 𝑣))
1918, 5elab2g 3683 . . . . . . . . 9 (𝐶𝐾 → (𝐶𝐾 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 𝑣))
2019ibi 267 . . . . . . . 8 (𝐶𝐾 → ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 𝑣)
21 frn 6744 . . . . . . . . . . . . . . 15 (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) → ran 𝑡 ⊆ (𝒫 𝑈 ∩ Fin))
2221ad2antrl 728 . . . . . . . . . . . . . 14 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ⊆ (𝒫 𝑈 ∩ Fin))
23 inss1 4245 . . . . . . . . . . . . . 14 (𝒫 𝑈 ∩ Fin) ⊆ 𝒫 𝑈
2422, 23sstrdi 4008 . . . . . . . . . . . . 13 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ⊆ 𝒫 𝑈)
25 sspwuni 5105 . . . . . . . . . . . . 13 (ran 𝑡 ⊆ 𝒫 𝑈 ran 𝑡𝑈)
2624, 25sylib 218 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡𝑈)
27 vex 3482 . . . . . . . . . . . . . . 15 𝑡 ∈ V
2827rnex 7933 . . . . . . . . . . . . . 14 ran 𝑡 ∈ V
2928uniex 7760 . . . . . . . . . . . . 13 ran 𝑡 ∈ V
3029elpw 4609 . . . . . . . . . . . 12 ( ran 𝑡 ∈ 𝒫 𝑈 ran 𝑡𝑈)
3126, 30sylibr 234 . . . . . . . . . . 11 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ∈ 𝒫 𝑈)
32 ffn 6737 . . . . . . . . . . . . . . 15 (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) → 𝑡 Fn 𝐴)
3332ad2antrl 728 . . . . . . . . . . . . . 14 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → 𝑡 Fn 𝐴)
34 dffn4 6827 . . . . . . . . . . . . . 14 (𝑡 Fn 𝐴𝑡:𝐴onto→ran 𝑡)
3533, 34sylib 218 . . . . . . . . . . . . 13 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → 𝑡:𝐴onto→ran 𝑡)
36 fofi 9349 . . . . . . . . . . . . 13 ((𝐴 ∈ Fin ∧ 𝑡:𝐴onto→ran 𝑡) → ran 𝑡 ∈ Fin)
3735, 36syldan 591 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ∈ Fin)
38 inss2 4246 . . . . . . . . . . . . 13 (𝒫 𝑈 ∩ Fin) ⊆ Fin
3922, 38sstrdi 4008 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ⊆ Fin)
40 unifi 9382 . . . . . . . . . . . 12 ((ran 𝑡 ∈ Fin ∧ ran 𝑡 ⊆ Fin) → ran 𝑡 ∈ Fin)
4137, 39, 40syl2anc 584 . . . . . . . . . . 11 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ∈ Fin)
4231, 41elind 4210 . . . . . . . . . 10 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ∈ (𝒫 𝑈 ∩ Fin))
4342adantlr 715 . . . . . . . . 9 (((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ∈ (𝒫 𝑈 ∩ Fin))
44 simplr 769 . . . . . . . . . 10 (((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → 𝐶 𝑥𝐴 𝐵)
45 fnfvelrn 7100 . . . . . . . . . . . . . . . . . 18 ((𝑡 Fn 𝐴𝑥𝐴) → (𝑡𝑥) ∈ ran 𝑡)
4632, 45sylan 580 . . . . . . . . . . . . . . . . 17 ((𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ 𝑥𝐴) → (𝑡𝑥) ∈ ran 𝑡)
4746adantll 714 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ Fin ∧ 𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin)) ∧ 𝑥𝐴) → (𝑡𝑥) ∈ ran 𝑡)
48 elssuni 4942 . . . . . . . . . . . . . . . 16 ((𝑡𝑥) ∈ ran 𝑡 → (𝑡𝑥) ⊆ ran 𝑡)
49 uniss 4920 . . . . . . . . . . . . . . . 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 4002 . . . . . . . . . . . . . . 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 3165 . . . . . . . . . . . . 13 ((𝐴 ∈ Fin ∧ 𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin)) → (∀𝑥𝐴 𝐵 (𝑡𝑥) → ∀𝑥𝐴 𝐵 ran 𝑡))
5453impr 454 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ∀𝑥𝐴 𝐵 ran 𝑡)
55 iunss 5050 . . . . . . . . . . . 12 ( 𝑥𝐴 𝐵 ran 𝑡 ↔ ∀𝑥𝐴 𝐵 ran 𝑡)
5654, 55sylibr 234 . . . . . . . . . . 11 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → 𝑥𝐴 𝐵 ran 𝑡)
5756adantlr 715 . . . . . . . . . 10 (((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → 𝑥𝐴 𝐵 ran 𝑡)
5844, 57sstrd 4006 . . . . . . . . 9 (((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → 𝐶 ran 𝑡)
59 unieq 4923 . . . . . . . . . . 11 (𝑣 = ran 𝑡 𝑣 = ran 𝑡)
6059sseq2d 4028 . . . . . . . . . 10 (𝑣 = ran 𝑡 → (𝐶 𝑣𝐶 ran 𝑡))
6160rspcev 3622 . . . . . . . . 9 (( ran 𝑡 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝐶 ran 𝑡) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 𝑣)
6243, 58, 61syl2anc 584 . . . . . . . 8 (((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 𝑣)
6320, 62nsyl3 138 . . . . . . 7 (((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ¬ 𝐶𝐾)
6463ex 412 . . . . . 6 ((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) → ((𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥)) → ¬ 𝐶𝐾))
6564exlimdv 1931 . . . . 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, 66biimtrid 242 . . 3 ((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) → (¬ ∃𝑥𝐴 𝐵𝐾 → ¬ 𝐶𝐾))
6867con4d 115 . 2 ((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) → (𝐶𝐾 → ∃𝑥𝐴 𝐵𝐾))
69683impia 1116 1 ((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵𝐶𝐾) → ∃𝑥𝐴 𝐵𝐾)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wral 3059  wrex 3068  Vcvv 3478  cin 3962  wss 3963  𝒫 cpw 4605   cuni 4912   ciun 4996  ran crn 5690   Fn wfn 6558  wf 6559  ontowfo 6561  cfv 6563  Fincfn 8984  MetOpencmopn 21372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1o 8505  df-en 8985  df-dom 8986  df-fin 8988
This theorem is referenced by:  heiborlem3  37800  heiborlem10  37807
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