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Theorem heiborlem1 34032
Description: Lemma for heibor 34042. 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 3786 . . . . . . . . . 10 (𝑢 = 𝐵 → (𝑢 𝑣𝐵 𝑣))
32rexbidv 3199 . . . . . . . . 9 (𝑢 = 𝐵 → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣))
43notbid 309 . . . . . . . 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 3509 . . . . . . 7 (𝐵𝐾 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣)
76con2bii 348 . . . . . 6 (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣 ↔ ¬ 𝐵𝐾)
87ralbii 3127 . . . . 5 (∀𝑥𝐴𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣 ↔ ∀𝑥𝐴 ¬ 𝐵𝐾)
9 ralnex 3139 . . . . 5 (∀𝑥𝐴 ¬ 𝐵𝐾 ↔ ¬ ∃𝑥𝐴 𝐵𝐾)
108, 9bitr2i 267 . . . 4 (¬ ∃𝑥𝐴 𝐵𝐾 ↔ ∀𝑥𝐴𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣)
11 unieq 4602 . . . . . . . . 9 (𝑣 = (𝑡𝑥) → 𝑣 = (𝑡𝑥))
1211sseq2d 3793 . . . . . . . 8 (𝑣 = (𝑡𝑥) → (𝐵 𝑣𝐵 (𝑡𝑥)))
1312ac6sfi 8411 . . . . . . 7 ((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣) → ∃𝑡(𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥)))
1413ex 401 . . . . . 6 (𝐴 ∈ Fin → (∀𝑥𝐴𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣 → ∃𝑡(𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))))
1514adantr 472 . . . . 5 ((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) → (∀𝑥𝐴𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣 → ∃𝑡(𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))))
16 sseq1 3786 . . . . . . . . . . . 12 (𝑢 = 𝐶 → (𝑢 𝑣𝐶 𝑣))
1716rexbidv 3199 . . . . . . . . . . 11 (𝑢 = 𝐶 → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 𝑣))
1817notbid 309 . . . . . . . . . 10 (𝑢 = 𝐶 → (¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 𝑣))
1918, 5elab2g 3508 . . . . . . . . 9 (𝐶𝐾 → (𝐶𝐾 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 𝑣))
2019ibi 258 . . . . . . . 8 (𝐶𝐾 → ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 𝑣)
21 frn 6229 . . . . . . . . . . . . . . 15 (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) → ran 𝑡 ⊆ (𝒫 𝑈 ∩ Fin))
2221ad2antrl 719 . . . . . . . . . . . . . 14 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ⊆ (𝒫 𝑈 ∩ Fin))
23 inss1 3992 . . . . . . . . . . . . . 14 (𝒫 𝑈 ∩ Fin) ⊆ 𝒫 𝑈
2422, 23syl6ss 3773 . . . . . . . . . . . . 13 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ⊆ 𝒫 𝑈)
25 sspwuni 4768 . . . . . . . . . . . . 13 (ran 𝑡 ⊆ 𝒫 𝑈 ran 𝑡𝑈)
2624, 25sylib 209 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡𝑈)
27 vex 3353 . . . . . . . . . . . . . . 15 𝑡 ∈ V
2827rnex 7298 . . . . . . . . . . . . . 14 ran 𝑡 ∈ V
2928uniex 7151 . . . . . . . . . . . . 13 ran 𝑡 ∈ V
3029elpw 4321 . . . . . . . . . . . 12 ( ran 𝑡 ∈ 𝒫 𝑈 ran 𝑡𝑈)
3126, 30sylibr 225 . . . . . . . . . . 11 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ∈ 𝒫 𝑈)
32 ffn 6223 . . . . . . . . . . . . . . 15 (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) → 𝑡 Fn 𝐴)
3332ad2antrl 719 . . . . . . . . . . . . . 14 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → 𝑡 Fn 𝐴)
34 dffn4 6304 . . . . . . . . . . . . . 14 (𝑡 Fn 𝐴𝑡:𝐴onto→ran 𝑡)
3533, 34sylib 209 . . . . . . . . . . . . 13 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → 𝑡:𝐴onto→ran 𝑡)
36 fofi 8459 . . . . . . . . . . . . 13 ((𝐴 ∈ Fin ∧ 𝑡:𝐴onto→ran 𝑡) → ran 𝑡 ∈ Fin)
3735, 36syldan 585 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ∈ Fin)
38 inss2 3993 . . . . . . . . . . . . 13 (𝒫 𝑈 ∩ Fin) ⊆ Fin
3922, 38syl6ss 3773 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ⊆ Fin)
40 unifi 8462 . . . . . . . . . . . 12 ((ran 𝑡 ∈ Fin ∧ ran 𝑡 ⊆ Fin) → ran 𝑡 ∈ Fin)
4137, 39, 40syl2anc 579 . . . . . . . . . . 11 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ∈ Fin)
4231, 41elind 3960 . . . . . . . . . 10 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ∈ (𝒫 𝑈 ∩ Fin))
4342adantlr 706 . . . . . . . . 9 (((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ∈ (𝒫 𝑈 ∩ Fin))
44 simplr 785 . . . . . . . . . 10 (((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → 𝐶 𝑥𝐴 𝐵)
45 fnfvelrn 6546 . . . . . . . . . . . . . . . . . 18 ((𝑡 Fn 𝐴𝑥𝐴) → (𝑡𝑥) ∈ ran 𝑡)
4632, 45sylan 575 . . . . . . . . . . . . . . . . 17 ((𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ 𝑥𝐴) → (𝑡𝑥) ∈ ran 𝑡)
4746adantll 705 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ Fin ∧ 𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin)) ∧ 𝑥𝐴) → (𝑡𝑥) ∈ ran 𝑡)
48 elssuni 4625 . . . . . . . . . . . . . . . 16 ((𝑡𝑥) ∈ ran 𝑡 → (𝑡𝑥) ⊆ ran 𝑡)
49 uniss 4617 . . . . . . . . . . . . . . . 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 3768 . . . . . . . . . . . . . . 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 3109 . . . . . . . . . . . . 13 ((𝐴 ∈ Fin ∧ 𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin)) → (∀𝑥𝐴 𝐵 (𝑡𝑥) → ∀𝑥𝐴 𝐵 ran 𝑡))
5453impr 446 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ∀𝑥𝐴 𝐵 ran 𝑡)
55 iunss 4717 . . . . . . . . . . . 12 ( 𝑥𝐴 𝐵 ran 𝑡 ↔ ∀𝑥𝐴 𝐵 ran 𝑡)
5654, 55sylibr 225 . . . . . . . . . . 11 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → 𝑥𝐴 𝐵 ran 𝑡)
5756adantlr 706 . . . . . . . . . 10 (((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → 𝑥𝐴 𝐵 ran 𝑡)
5844, 57sstrd 3771 . . . . . . . . 9 (((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → 𝐶 ran 𝑡)
59 unieq 4602 . . . . . . . . . . 11 (𝑣 = ran 𝑡 𝑣 = ran 𝑡)
6059sseq2d 3793 . . . . . . . . . 10 (𝑣 = ran 𝑡 → (𝐶 𝑣𝐶 ran 𝑡))
6160rspcev 3461 . . . . . . . . 9 (( ran 𝑡 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝐶 ran 𝑡) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 𝑣)
6243, 58, 61syl2anc 579 . . . . . . . 8 (((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 𝑣)
6320, 62nsyl3 135 . . . . . . 7 (((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ¬ 𝐶𝐾)
6463ex 401 . . . . . 6 ((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) → ((𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥)) → ¬ 𝐶𝐾))
6564exlimdv 2028 . . . . 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 233 . . 3 ((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) → (¬ ∃𝑥𝐴 𝐵𝐾 → ¬ 𝐶𝐾))
6867con4d 115 . 2 ((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) → (𝐶𝐾 → ∃𝑥𝐴 𝐵𝐾))
69683impia 1145 1 ((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵𝐶𝐾) → ∃𝑥𝐴 𝐵𝐾)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1107   = wceq 1652  wex 1874  wcel 2155  {cab 2751  wral 3055  wrex 3056  Vcvv 3350  cin 3731  wss 3732  𝒫 cpw 4315   cuni 4594   ciun 4676  ran crn 5278   Fn wfn 6063  wf 6064  ontowfo 6066  cfv 6068  Fincfn 8160  MetOpencmopn 20009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-en 8161  df-dom 8162  df-fin 8164
This theorem is referenced by:  heiborlem3  34034  heiborlem10  34041
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