MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fin23lem21 Structured version   Visualization version   GIF version

Theorem fin23lem21 9105
Description: Lemma for fin23 9155. 𝑋 is not empty. We only need here that 𝑡 has at least one set in its range besides ; the much stronger hypothesis here will serve as our induction hypothesis though. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by Mario Carneiro, 6-May-2015.)
Hypotheses
Ref Expression
fin23lem.a 𝑈 = seq𝜔((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)
fin23lem17.f 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔𝑚 ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
Assertion
Ref Expression
fin23lem21 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → ran 𝑈 ≠ ∅)
Distinct variable groups:   𝑔,𝑖,𝑡,𝑢,𝑥,𝑎   𝐹,𝑎,𝑡   𝑉,𝑎   𝑥,𝑎   𝑈,𝑎,𝑖,𝑢   𝑔,𝑎
Allowed substitution hints:   𝑈(𝑥,𝑡,𝑔)   𝐹(𝑥,𝑢,𝑔,𝑖)   𝑉(𝑥,𝑢,𝑡,𝑔,𝑖)

Proof of Theorem fin23lem21
StepHypRef Expression
1 fin23lem.a . . 3 𝑈 = seq𝜔((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)
2 fin23lem17.f . . 3 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔𝑚 ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
31, 2fin23lem17 9104 . 2 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → ran 𝑈 ∈ ran 𝑈)
41fnseqom 7495 . . . . 5 𝑈 Fn ω
5 fvelrnb 6200 . . . . 5 (𝑈 Fn ω → ( ran 𝑈 ∈ ran 𝑈 ↔ ∃𝑎 ∈ ω (𝑈𝑎) = ran 𝑈))
64, 5ax-mp 5 . . . 4 ( ran 𝑈 ∈ ran 𝑈 ↔ ∃𝑎 ∈ ω (𝑈𝑎) = ran 𝑈)
7 id 22 . . . . . . 7 (𝑎 ∈ ω → 𝑎 ∈ ω)
8 vex 3189 . . . . . . . . . 10 𝑡 ∈ V
9 f1f1orn 6105 . . . . . . . . . 10 (𝑡:ω–1-1𝑉𝑡:ω–1-1-onto→ran 𝑡)
10 f1oen3g 7915 . . . . . . . . . 10 ((𝑡 ∈ V ∧ 𝑡:ω–1-1-onto→ran 𝑡) → ω ≈ ran 𝑡)
118, 9, 10sylancr 694 . . . . . . . . 9 (𝑡:ω–1-1𝑉 → ω ≈ ran 𝑡)
12 ominf 8116 . . . . . . . . 9 ¬ ω ∈ Fin
13 ssdif0 3916 . . . . . . . . . . 11 (ran 𝑡 ⊆ {∅} ↔ (ran 𝑡 ∖ {∅}) = ∅)
14 snfi 7982 . . . . . . . . . . . . 13 {∅} ∈ Fin
15 ssfi 8124 . . . . . . . . . . . . 13 (({∅} ∈ Fin ∧ ran 𝑡 ⊆ {∅}) → ran 𝑡 ∈ Fin)
1614, 15mpan 705 . . . . . . . . . . . 12 (ran 𝑡 ⊆ {∅} → ran 𝑡 ∈ Fin)
17 enfi 8120 . . . . . . . . . . . 12 (ω ≈ ran 𝑡 → (ω ∈ Fin ↔ ran 𝑡 ∈ Fin))
1816, 17syl5ibr 236 . . . . . . . . . . 11 (ω ≈ ran 𝑡 → (ran 𝑡 ⊆ {∅} → ω ∈ Fin))
1913, 18syl5bir 233 . . . . . . . . . 10 (ω ≈ ran 𝑡 → ((ran 𝑡 ∖ {∅}) = ∅ → ω ∈ Fin))
2019necon3bd 2804 . . . . . . . . 9 (ω ≈ ran 𝑡 → (¬ ω ∈ Fin → (ran 𝑡 ∖ {∅}) ≠ ∅))
2111, 12, 20mpisyl 21 . . . . . . . 8 (𝑡:ω–1-1𝑉 → (ran 𝑡 ∖ {∅}) ≠ ∅)
22 n0 3907 . . . . . . . . 9 ((ran 𝑡 ∖ {∅}) ≠ ∅ ↔ ∃𝑎 𝑎 ∈ (ran 𝑡 ∖ {∅}))
23 eldifsn 4287 . . . . . . . . . . 11 (𝑎 ∈ (ran 𝑡 ∖ {∅}) ↔ (𝑎 ∈ ran 𝑡𝑎 ≠ ∅))
24 elssuni 4433 . . . . . . . . . . . 12 (𝑎 ∈ ran 𝑡𝑎 ran 𝑡)
25 ssn0 3948 . . . . . . . . . . . 12 ((𝑎 ran 𝑡𝑎 ≠ ∅) → ran 𝑡 ≠ ∅)
2624, 25sylan 488 . . . . . . . . . . 11 ((𝑎 ∈ ran 𝑡𝑎 ≠ ∅) → ran 𝑡 ≠ ∅)
2723, 26sylbi 207 . . . . . . . . . 10 (𝑎 ∈ (ran 𝑡 ∖ {∅}) → ran 𝑡 ≠ ∅)
2827exlimiv 1855 . . . . . . . . 9 (∃𝑎 𝑎 ∈ (ran 𝑡 ∖ {∅}) → ran 𝑡 ≠ ∅)
2922, 28sylbi 207 . . . . . . . 8 ((ran 𝑡 ∖ {∅}) ≠ ∅ → ran 𝑡 ≠ ∅)
3021, 29syl 17 . . . . . . 7 (𝑡:ω–1-1𝑉 ran 𝑡 ≠ ∅)
311fin23lem14 9099 . . . . . . 7 ((𝑎 ∈ ω ∧ ran 𝑡 ≠ ∅) → (𝑈𝑎) ≠ ∅)
327, 30, 31syl2anr 495 . . . . . 6 ((𝑡:ω–1-1𝑉𝑎 ∈ ω) → (𝑈𝑎) ≠ ∅)
33 neeq1 2852 . . . . . 6 ((𝑈𝑎) = ran 𝑈 → ((𝑈𝑎) ≠ ∅ ↔ ran 𝑈 ≠ ∅))
3432, 33syl5ibcom 235 . . . . 5 ((𝑡:ω–1-1𝑉𝑎 ∈ ω) → ((𝑈𝑎) = ran 𝑈 ran 𝑈 ≠ ∅))
3534rexlimdva 3024 . . . 4 (𝑡:ω–1-1𝑉 → (∃𝑎 ∈ ω (𝑈𝑎) = ran 𝑈 ran 𝑈 ≠ ∅))
366, 35syl5bi 232 . . 3 (𝑡:ω–1-1𝑉 → ( ran 𝑈 ∈ ran 𝑈 ran 𝑈 ≠ ∅))
3736adantl 482 . 2 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → ( ran 𝑈 ∈ ran 𝑈 ran 𝑈 ≠ ∅))
383, 37mpd 15 1 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → ran 𝑈 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wne 2790  wral 2907  wrex 2908  Vcvv 3186  cdif 3552  cin 3554  wss 3555  c0 3891  ifcif 4058  𝒫 cpw 4130  {csn 4148   cuni 4402   cint 4440   class class class wbr 4613  ran crn 5075  suc csuc 5684   Fn wfn 5842  1-1wf1 5844  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604  cmpt2 6606  ωcom 7012  seq𝜔cseqom 7487  𝑚 cmap 7802  cen 7896  Fincfn 7899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-seqom 7488  df-1o 7505  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903
This theorem is referenced by:  fin23lem31  9109
  Copyright terms: Public domain W3C validator