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Theorem heiborlem1 33942
Description: Lemma for heibor 33952. 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 3775 . . . . . . . . . 10 (𝑢 = 𝐵 → (𝑢 𝑣𝐵 𝑣))
32rexbidv 3200 . . . . . . . . 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 3505 . . . . . . 7 (𝐵𝐾 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣)
76con2bii 346 . . . . . 6 (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣 ↔ ¬ 𝐵𝐾)
87ralbii 3129 . . . . 5 (∀𝑥𝐴𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣 ↔ ∀𝑥𝐴 ¬ 𝐵𝐾)
9 ralnex 3141 . . . . 5 (∀𝑥𝐴 ¬ 𝐵𝐾 ↔ ¬ ∃𝑥𝐴 𝐵𝐾)
108, 9bitr2i 265 . . . 4 (¬ ∃𝑥𝐴 𝐵𝐾 ↔ ∀𝑥𝐴𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣)
11 unieq 4582 . . . . . . . . 9 (𝑣 = (𝑡𝑥) → 𝑣 = (𝑡𝑥))
1211sseq2d 3782 . . . . . . . 8 (𝑣 = (𝑡𝑥) → (𝐵 𝑣𝐵 (𝑡𝑥)))
1312ac6sfi 8360 . . . . . . 7 ((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣) → ∃𝑡(𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥)))
1413ex 397 . . . . . 6 (𝐴 ∈ Fin → (∀𝑥𝐴𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣 → ∃𝑡(𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))))
1514adantr 466 . . . . 5 ((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) → (∀𝑥𝐴𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 𝑣 → ∃𝑡(𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))))
16 sseq1 3775 . . . . . . . . . . . 12 (𝑢 = 𝐶 → (𝑢 𝑣𝐶 𝑣))
1716rexbidv 3200 . . . . . . . . . . 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 3504 . . . . . . . . 9 (𝐶𝐾 → (𝐶𝐾 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 𝑣))
2019ibi 256 . . . . . . . 8 (𝐶𝐾 → ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 𝑣)
21 frn 6193 . . . . . . . . . . . . . . 15 (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) → ran 𝑡 ⊆ (𝒫 𝑈 ∩ Fin))
2221ad2antrl 707 . . . . . . . . . . . . . 14 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ⊆ (𝒫 𝑈 ∩ Fin))
23 inss1 3981 . . . . . . . . . . . . . 14 (𝒫 𝑈 ∩ Fin) ⊆ 𝒫 𝑈
2422, 23syl6ss 3764 . . . . . . . . . . . . 13 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ⊆ 𝒫 𝑈)
25 sspwuni 4745 . . . . . . . . . . . . 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 3354 . . . . . . . . . . . . . . 15 𝑡 ∈ V
2827rnex 7247 . . . . . . . . . . . . . 14 ran 𝑡 ∈ V
2928uniex 7100 . . . . . . . . . . . . 13 ran 𝑡 ∈ V
3029elpw 4303 . . . . . . . . . . . 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 6185 . . . . . . . . . . . . . . 15 (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) → 𝑡 Fn 𝐴)
3332ad2antrl 707 . . . . . . . . . . . . . 14 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → 𝑡 Fn 𝐴)
34 dffn4 6262 . . . . . . . . . . . . . 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 8408 . . . . . . . . . . . . 13 ((𝐴 ∈ Fin ∧ 𝑡:𝐴onto→ran 𝑡) → ran 𝑡 ∈ Fin)
3735, 36syldan 579 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ∈ Fin)
38 inss2 3982 . . . . . . . . . . . . 13 (𝒫 𝑈 ∩ Fin) ⊆ Fin
3922, 38syl6ss 3764 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ⊆ Fin)
40 unifi 8411 . . . . . . . . . . . 12 ((ran 𝑡 ∈ Fin ∧ ran 𝑡 ⊆ Fin) → ran 𝑡 ∈ Fin)
4137, 39, 40syl2anc 573 . . . . . . . . . . 11 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ∈ Fin)
4231, 41elind 3949 . . . . . . . . . 10 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ∈ (𝒫 𝑈 ∩ Fin))
4342adantlr 694 . . . . . . . . 9 (((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ran 𝑡 ∈ (𝒫 𝑈 ∩ Fin))
44 simplr 752 . . . . . . . . . 10 (((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → 𝐶 𝑥𝐴 𝐵)
45 fnfvelrn 6499 . . . . . . . . . . . . . . . . . 18 ((𝑡 Fn 𝐴𝑥𝐴) → (𝑡𝑥) ∈ ran 𝑡)
4632, 45sylan 569 . . . . . . . . . . . . . . . . 17 ((𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ 𝑥𝐴) → (𝑡𝑥) ∈ ran 𝑡)
4746adantll 693 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ Fin ∧ 𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin)) ∧ 𝑥𝐴) → (𝑡𝑥) ∈ ran 𝑡)
48 elssuni 4603 . . . . . . . . . . . . . . . 16 ((𝑡𝑥) ∈ ran 𝑡 → (𝑡𝑥) ⊆ ran 𝑡)
49 uniss 4595 . . . . . . . . . . . . . . . 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 3759 . . . . . . . . . . . . . . 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 3111 . . . . . . . . . . . . 13 ((𝐴 ∈ Fin ∧ 𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin)) → (∀𝑥𝐴 𝐵 (𝑡𝑥) → ∀𝑥𝐴 𝐵 ran 𝑡))
5453impr 442 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → ∀𝑥𝐴 𝐵 ran 𝑡)
55 iunss 4695 . . . . . . . . . . . 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 694 . . . . . . . . . 10 (((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → 𝑥𝐴 𝐵 ran 𝑡)
5844, 57sstrd 3762 . . . . . . . . 9 (((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥))) → 𝐶 ran 𝑡)
59 unieq 4582 . . . . . . . . . . 11 (𝑣 = ran 𝑡 𝑣 = ran 𝑡)
6059sseq2d 3782 . . . . . . . . . 10 (𝑣 = ran 𝑡 → (𝐶 𝑣𝐶 ran 𝑡))
6160rspcev 3460 . . . . . . . . 9 (( ran 𝑡 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝐶 ran 𝑡) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 𝑣)
6243, 58, 61syl2anc 573 . . . . . . . 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 397 . . . . . 6 ((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) → ((𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥𝐴 𝐵 (𝑡𝑥)) → ¬ 𝐶𝐾))
6564exlimdv 2013 . . . . 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 115 . 2 ((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵) → (𝐶𝐾 → ∃𝑥𝐴 𝐵𝐾))
69683impia 1109 1 ((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵𝐶𝐾) → ∃𝑥𝐴 𝐵𝐾)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  w3a 1071   = wceq 1631  wex 1852  wcel 2145  {cab 2757  wral 3061  wrex 3062  Vcvv 3351  cin 3722  wss 3723  𝒫 cpw 4297   cuni 4574   ciun 4654  ran crn 5250   Fn wfn 6026  wf 6027  ontowfo 6029  cfv 6031  Fincfn 8109  MetOpencmopn 19951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-en 8110  df-dom 8111  df-fin 8113
This theorem is referenced by:  heiborlem3  33944  heiborlem10  33951
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