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Theorem fin23lem21 9199
 Description: Lemma for fin23 9249. 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 9198 . 2 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → ran 𝑈 ∈ ran 𝑈)
41fnseqom 7595 . . . . 5 𝑈 Fn ω
5 fvelrnb 6282 . . . . 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 3234 . . . . . . . . . 10 𝑡 ∈ V
9 f1f1orn 6186 . . . . . . . . . 10 (𝑡:ω–1-1𝑉𝑡:ω–1-1-onto→ran 𝑡)
10 f1oen3g 8013 . . . . . . . . . 10 ((𝑡 ∈ V ∧ 𝑡:ω–1-1-onto→ran 𝑡) → ω ≈ ran 𝑡)
118, 9, 10sylancr 696 . . . . . . . . 9 (𝑡:ω–1-1𝑉 → ω ≈ ran 𝑡)
12 ominf 8213 . . . . . . . . 9 ¬ ω ∈ Fin
13 ssdif0 3975 . . . . . . . . . . 11 (ran 𝑡 ⊆ {∅} ↔ (ran 𝑡 ∖ {∅}) = ∅)
14 snfi 8079 . . . . . . . . . . . . 13 {∅} ∈ Fin
15 ssfi 8221 . . . . . . . . . . . . 13 (({∅} ∈ Fin ∧ ran 𝑡 ⊆ {∅}) → ran 𝑡 ∈ Fin)
1614, 15mpan 706 . . . . . . . . . . . 12 (ran 𝑡 ⊆ {∅} → ran 𝑡 ∈ Fin)
17 enfi 8217 . . . . . . . . . . . 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 2837 . . . . . . . . 9 (ω ≈ ran 𝑡 → (¬ ω ∈ Fin → (ran 𝑡 ∖ {∅}) ≠ ∅))
2111, 12, 20mpisyl 21 . . . . . . . 8 (𝑡:ω–1-1𝑉 → (ran 𝑡 ∖ {∅}) ≠ ∅)
22 n0 3964 . . . . . . . . 9 ((ran 𝑡 ∖ {∅}) ≠ ∅ ↔ ∃𝑎 𝑎 ∈ (ran 𝑡 ∖ {∅}))
23 eldifsn 4350 . . . . . . . . . . 11 (𝑎 ∈ (ran 𝑡 ∖ {∅}) ↔ (𝑎 ∈ ran 𝑡𝑎 ≠ ∅))
24 elssuni 4499 . . . . . . . . . . . 12 (𝑎 ∈ ran 𝑡𝑎 ran 𝑡)
25 ssn0 4009 . . . . . . . . . . . 12 ((𝑎 ran 𝑡𝑎 ≠ ∅) → ran 𝑡 ≠ ∅)
2624, 25sylan 487 . . . . . . . . . . 11 ((𝑎 ∈ ran 𝑡𝑎 ≠ ∅) → ran 𝑡 ≠ ∅)
2723, 26sylbi 207 . . . . . . . . . 10 (𝑎 ∈ (ran 𝑡 ∖ {∅}) → ran 𝑡 ≠ ∅)
2827exlimiv 1898 . . . . . . . . 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 9193 . . . . . . 7 ((𝑎 ∈ ω ∧ ran 𝑡 ≠ ∅) → (𝑈𝑎) ≠ ∅)
327, 30, 31syl2anr 494 . . . . . 6 ((𝑡:ω–1-1𝑉𝑎 ∈ ω) → (𝑈𝑎) ≠ ∅)
33 neeq1 2885 . . . . . 6 ((𝑈𝑎) = ran 𝑈 → ((𝑈𝑎) ≠ ∅ ↔ ran 𝑈 ≠ ∅))
3432, 33syl5ibcom 235 . . . . 5 ((𝑡:ω–1-1𝑉𝑎 ∈ ω) → ((𝑈𝑎) = ran 𝑈 ran 𝑈 ≠ ∅))
3534rexlimdva 3060 . . . 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 481 . 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 383   = wceq 1523  ∃wex 1744   ∈ wcel 2030  {cab 2637   ≠ wne 2823  ∀wral 2941  ∃wrex 2942  Vcvv 3231   ∖ cdif 3604   ∩ cin 3606   ⊆ wss 3607  ∅c0 3948  ifcif 4119  𝒫 cpw 4191  {csn 4210  ∪ cuni 4468  ∩ cint 4507   class class class wbr 4685  ran crn 5144  suc csuc 5763   Fn wfn 5921  –1-1→wf1 5923  –1-1-onto→wf1o 5925  ‘cfv 5926  (class class class)co 6690   ↦ cmpt2 6692  ωcom 7107  seq𝜔cseqom 7587   ↑𝑚 cmap 7899   ≈ cen 7994  Fincfn 7997 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-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-seqom 7588  df-1o 7605  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001 This theorem is referenced by:  fin23lem31  9203
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