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Theorem heiborlem1 33740
Description: Lemma for heibor 33750. 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 3659 . . . . . . . . . 10 (𝑢 = 𝐵 → (𝑢 𝑣𝐵 𝑣))
32rexbidv 3081 . . . . . . . . 9 (𝑢 = 𝐵 → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣))
43notbid 307 . . . . . . . 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 3386 . . . . . . 7 (𝐵𝐾 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣)
76con2bii 346 . . . . . 6 (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣 ↔ ¬ 𝐵𝐾)
87ralbii 3009 . . . . 5 (∀𝑥𝐴𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣 ↔ ∀𝑥𝐴 ¬ 𝐵𝐾)
9 ralnex 3021 . . . . 5 (∀𝑥𝐴 ¬ 𝐵𝐾 ↔ ¬ ∃𝑥𝐴 𝐵𝐾)
108, 9bitr2i 265 . . . 4 (¬ ∃𝑥𝐴 𝐵𝐾 ↔ ∀𝑥𝐴𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣)
11 unieq 4476 . . . . . . . . 9 (𝑣 = (𝑡𝑥) → 𝑣 = (𝑡𝑥))
1211sseq2d 3666 . . . . . . . 8 (𝑣 = (𝑡𝑥) → (𝐵 𝑣𝐵 (𝑡𝑥)))
1312ac6sfi 8245 . . . . . . 7 ((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣) → ∃𝑡(𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥)))
1413ex 449 . . . . . 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 3659 . . . . . . . . . . . 12 (𝑢 = 𝐶 → (𝑢 𝑣𝐶 𝑣))
1716rexbidv 3081 . . . . . . . . . . 11 (𝑢 = 𝐶 → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 𝑣))
1817notbid 307 . . . . . . . . . 10 (𝑢 = 𝐶 → (¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 𝑣))
1918, 5elab2g 3385 . . . . . . . . 9 (𝐶𝐾 → (𝐶𝐾 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 𝑣))
2019ibi 256 . . . . . . . 8 (𝐶𝐾 → ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 𝑣)
21 frn 6091 . . . . . . . . . . . . . . 15 (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) → ran 𝑡 ⊆ (𝒫 𝑈 ∩ Fin))
2221ad2antrl 764 . . . . . . . . . . . . . 14 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ⊆ (𝒫 𝑈 ∩ Fin))
23 inss1 3866 . . . . . . . . . . . . . 14 (𝒫 𝑈 ∩ Fin) ⊆ 𝒫 𝑈
2422, 23syl6ss 3648 . . . . . . . . . . . . 13 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ⊆ 𝒫 𝑈)
25 sspwuni 4643 . . . . . . . . . . . . 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 3234 . . . . . . . . . . . . . . 15 𝑡 ∈ V
2827rnex 7142 . . . . . . . . . . . . . 14 ran 𝑡 ∈ V
2928uniex 6995 . . . . . . . . . . . . 13 ran 𝑡 ∈ V
3029elpw 4197 . . . . . . . . . . . 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 6083 . . . . . . . . . . . . . . 15 (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) → 𝑡 Fn 𝐴)
3332ad2antrl 764 . . . . . . . . . . . . . 14 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → 𝑡 Fn 𝐴)
34 dffn4 6159 . . . . . . . . . . . . . 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 8293 . . . . . . . . . . . . 13 ((𝐴 ∈ Fin ∧ 𝑡:𝐴onto→ran 𝑡) → ran 𝑡 ∈ Fin)
3735, 36syldan 486 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ∈ Fin)
38 inss2 3867 . . . . . . . . . . . . 13 (𝒫 𝑈 ∩ Fin) ⊆ Fin
3922, 38syl6ss 3648 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ⊆ Fin)
40 unifi 8296 . . . . . . . . . . . 12 ((ran 𝑡 ∈ Fin ∧ ran 𝑡 ⊆ Fin) → ran 𝑡 ∈ Fin)
4137, 39, 40syl2anc 694 . . . . . . . . . . 11 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ∈ Fin)
4231, 41elind 3831 . . . . . . . . . 10 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ∈ (𝒫 𝑈 ∩ Fin))
4342adantlr 751 . . . . . . . . 9 (((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ∈ (𝒫 𝑈 ∩ Fin))
44 simplr 807 . . . . . . . . . 10 (((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → 𝐶 𝑥𝐴 𝐵)
45 fnfvelrn 6396 . . . . . . . . . . . . . . . . . 18 ((𝑡 Fn 𝐴𝑥𝐴) → (𝑡𝑥) ∈ ran 𝑡)
4632, 45sylan 487 . . . . . . . . . . . . . . . . 17 ((𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ 𝑥𝐴) → (𝑡𝑥) ∈ ran 𝑡)
4746adantll 750 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ Fin ∧ 𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin)) ∧ 𝑥𝐴) → (𝑡𝑥) ∈ ran 𝑡)
48 elssuni 4499 . . . . . . . . . . . . . . . 16 ((𝑡𝑥) ∈ ran 𝑡 → (𝑡𝑥) ⊆ ran 𝑡)
49 uniss 4490 . . . . . . . . . . . . . . . 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 3643 . . . . . . . . . . . . . . 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 2991 . . . . . . . . . . . . 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 4593 . . . . . . . . . . . 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 751 . . . . . . . . . 10 (((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → 𝑥𝐴 𝐵 ran 𝑡)
5844, 57sstrd 3646 . . . . . . . . 9 (((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → 𝐶 ran 𝑡)
59 unieq 4476 . . . . . . . . . . 11 (𝑣 = ran 𝑡 𝑣 = ran 𝑡)
6059sseq2d 3666 . . . . . . . . . 10 (𝑣 = ran 𝑡 → (𝐶 𝑣𝐶 ran 𝑡))
6160rspcev 3340 . . . . . . . . 9 (( ran 𝑡 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝐶 ran 𝑡) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 𝑣)
6243, 58, 61syl2anc 694 . . . . . . . 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 449 . . . . . 6 ((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) → ((𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥)) → ¬ 𝐶𝐾))
6564exlimdv 1901 . . . . 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 1280 1 ((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵𝐶𝐾) → ∃𝑥𝐴 𝐵𝐾)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wral 2941  wrex 2942  Vcvv 3231  cin 3606  wss 3607  𝒫 cpw 4191   cuni 4468   ciun 4552  ran crn 5144   Fn wfn 5921  wf 5922  ontowfo 5924  cfv 5926  Fincfn 7997  MetOpencmopn 19784
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-3or 1055  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-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-fin 8001
This theorem is referenced by:  heiborlem3  33742  heiborlem10  33749
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