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

Theorem fin23lem21 10382
Description: Lemma for fin23 10432. 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 10381 . 2 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → ran 𝑈 ∈ ran 𝑈)
41fnseqom 8485 . . . . 5 𝑈 Fn ω
5 fvelrnb 6963 . . . . 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 3466 . . . . . . . . . 10 𝑡 ∈ V
9 f1f1orn 6854 . . . . . . . . . 10 (𝑡:ω–1-1𝑉𝑡:ω–1-1-onto→ran 𝑡)
10 f1oen3g 8997 . . . . . . . . . 10 ((𝑡 ∈ V ∧ 𝑡:ω–1-1-onto→ran 𝑡) → ω ≈ ran 𝑡)
118, 9, 10sylancr 585 . . . . . . . . 9 (𝑡:ω–1-1𝑉 → ω ≈ ran 𝑡)
12 ominf 9292 . . . . . . . . 9 ¬ ω ∈ Fin
13 ssdif0 4366 . . . . . . . . . . 11 (ran 𝑡 ⊆ {∅} ↔ (ran 𝑡 ∖ {∅}) = ∅)
14 snfi 9081 . . . . . . . . . . . . 13 {∅} ∈ Fin
15 ssfi 9211 . . . . . . . . . . . . 13 (({∅} ∈ Fin ∧ ran 𝑡 ⊆ {∅}) → ran 𝑡 ∈ Fin)
1614, 15mpan 688 . . . . . . . . . . . 12 (ran 𝑡 ⊆ {∅} → ran 𝑡 ∈ Fin)
17 enfi 9224 . . . . . . . . . . . 12 (ω ≈ ran 𝑡 → (ω ∈ Fin ↔ ran 𝑡 ∈ Fin))
1816, 17imbitrrid 245 . . . . . . . . . . 11 (ω ≈ ran 𝑡 → (ran 𝑡 ⊆ {∅} → ω ∈ Fin))
1913, 18biimtrrid 242 . . . . . . . . . 10 (ω ≈ ran 𝑡 → ((ran 𝑡 ∖ {∅}) = ∅ → ω ∈ Fin))
2019necon3bd 2944 . . . . . . . . 9 (ω ≈ ran 𝑡 → (¬ ω ∈ Fin → (ran 𝑡 ∖ {∅}) ≠ ∅))
2111, 12, 20mpisyl 21 . . . . . . . 8 (𝑡:ω–1-1𝑉 → (ran 𝑡 ∖ {∅}) ≠ ∅)
22 n0 4349 . . . . . . . . 9 ((ran 𝑡 ∖ {∅}) ≠ ∅ ↔ ∃𝑎 𝑎 ∈ (ran 𝑡 ∖ {∅}))
23 eldifsn 4795 . . . . . . . . . . 11 (𝑎 ∈ (ran 𝑡 ∖ {∅}) ↔ (𝑎 ∈ ran 𝑡𝑎 ≠ ∅))
24 elssuni 4945 . . . . . . . . . . . 12 (𝑎 ∈ ran 𝑡𝑎 ran 𝑡)
25 ssn0 4405 . . . . . . . . . . . 12 ((𝑎 ran 𝑡𝑎 ≠ ∅) → ran 𝑡 ≠ ∅)
2624, 25sylan 578 . . . . . . . . . . 11 ((𝑎 ∈ ran 𝑡𝑎 ≠ ∅) → ran 𝑡 ≠ ∅)
2723, 26sylbi 216 . . . . . . . . . 10 (𝑎 ∈ (ran 𝑡 ∖ {∅}) → ran 𝑡 ≠ ∅)
2827exlimiv 1926 . . . . . . . . 9 (∃𝑎 𝑎 ∈ (ran 𝑡 ∖ {∅}) → ran 𝑡 ≠ ∅)
2922, 28sylbi 216 . . . . . . . 8 ((ran 𝑡 ∖ {∅}) ≠ ∅ → ran 𝑡 ≠ ∅)
3021, 29syl 17 . . . . . . 7 (𝑡:ω–1-1𝑉 ran 𝑡 ≠ ∅)
311fin23lem14 10376 . . . . . . 7 ((𝑎 ∈ ω ∧ ran 𝑡 ≠ ∅) → (𝑈𝑎) ≠ ∅)
327, 30, 31syl2anr 595 . . . . . 6 ((𝑡:ω–1-1𝑉𝑎 ∈ ω) → (𝑈𝑎) ≠ ∅)
33 neeq1 2993 . . . . . 6 ((𝑈𝑎) = ran 𝑈 → ((𝑈𝑎) ≠ ∅ ↔ ran 𝑈 ≠ ∅))
3432, 33syl5ibcom 244 . . . . 5 ((𝑡:ω–1-1𝑉𝑎 ∈ ω) → ((𝑈𝑎) = ran 𝑈 ran 𝑈 ≠ ∅))
3534rexlimdva 3145 . . . 4 (𝑡:ω–1-1𝑉 → (∃𝑎 ∈ ω (𝑈𝑎) = ran 𝑈 ran 𝑈 ≠ ∅))
366, 35biimtrid 241 . . 3 (𝑡:ω–1-1𝑉 → ( ran 𝑈 ∈ ran 𝑈 ran 𝑈 ≠ ∅))
3736adantl 480 . 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 205  wa 394   = wceq 1534  wex 1774  wcel 2099  {cab 2703  wne 2930  wral 3051  wrex 3060  Vcvv 3462  cdif 3944  cin 3946  wss 3947  c0 4325  ifcif 4533  𝒫 cpw 4607  {csn 4633   cuni 4913   cint 4954   class class class wbr 5153  ran crn 5683  suc csuc 6378   Fn wfn 6549  1-1wf1 6551  1-1-ontowf1o 6553  cfv 6554  (class class class)co 7424  cmpo 7426  ωcom 7876  seqωcseqom 8477  m cmap 8855  cen 8971  Fincfn 8974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-seqom 8478  df-1o 8496  df-map 8857  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978
This theorem is referenced by:  fin23lem31  10386
  Copyright terms: Public domain W3C validator