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Theorem fin23lem21 9754
Description: Lemma for fin23 9804. 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 9753 . 2 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → ran 𝑈 ∈ ran 𝑈)
41fnseqom 8078 . . . . 5 𝑈 Fn ω
5 fvelrnb 6705 . . . . 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 3447 . . . . . . . . . 10 𝑡 ∈ V
9 f1f1orn 6605 . . . . . . . . . 10 (𝑡:ω–1-1𝑉𝑡:ω–1-1-onto→ran 𝑡)
10 f1oen3g 8512 . . . . . . . . . 10 ((𝑡 ∈ V ∧ 𝑡:ω–1-1-onto→ran 𝑡) → ω ≈ ran 𝑡)
118, 9, 10sylancr 590 . . . . . . . . 9 (𝑡:ω–1-1𝑉 → ω ≈ ran 𝑡)
12 ominf 8718 . . . . . . . . 9 ¬ ω ∈ Fin
13 ssdif0 4280 . . . . . . . . . . 11 (ran 𝑡 ⊆ {∅} ↔ (ran 𝑡 ∖ {∅}) = ∅)
14 snfi 8581 . . . . . . . . . . . . 13 {∅} ∈ Fin
15 ssfi 8726 . . . . . . . . . . . . 13 (({∅} ∈ Fin ∧ ran 𝑡 ⊆ {∅}) → ran 𝑡 ∈ Fin)
1614, 15mpan 689 . . . . . . . . . . . 12 (ran 𝑡 ⊆ {∅} → ran 𝑡 ∈ Fin)
17 enfi 8722 . . . . . . . . . . . 12 (ω ≈ ran 𝑡 → (ω ∈ Fin ↔ ran 𝑡 ∈ Fin))
1816, 17syl5ibr 249 . . . . . . . . . . 11 (ω ≈ ran 𝑡 → (ran 𝑡 ⊆ {∅} → ω ∈ Fin))
1913, 18syl5bir 246 . . . . . . . . . 10 (ω ≈ ran 𝑡 → ((ran 𝑡 ∖ {∅}) = ∅ → ω ∈ Fin))
2019necon3bd 3004 . . . . . . . . 9 (ω ≈ ran 𝑡 → (¬ ω ∈ Fin → (ran 𝑡 ∖ {∅}) ≠ ∅))
2111, 12, 20mpisyl 21 . . . . . . . 8 (𝑡:ω–1-1𝑉 → (ran 𝑡 ∖ {∅}) ≠ ∅)
22 n0 4263 . . . . . . . . 9 ((ran 𝑡 ∖ {∅}) ≠ ∅ ↔ ∃𝑎 𝑎 ∈ (ran 𝑡 ∖ {∅}))
23 eldifsn 4683 . . . . . . . . . . 11 (𝑎 ∈ (ran 𝑡 ∖ {∅}) ↔ (𝑎 ∈ ran 𝑡𝑎 ≠ ∅))
24 elssuni 4833 . . . . . . . . . . . 12 (𝑎 ∈ ran 𝑡𝑎 ran 𝑡)
25 ssn0 4311 . . . . . . . . . . . 12 ((𝑎 ran 𝑡𝑎 ≠ ∅) → ran 𝑡 ≠ ∅)
2624, 25sylan 583 . . . . . . . . . . 11 ((𝑎 ∈ ran 𝑡𝑎 ≠ ∅) → ran 𝑡 ≠ ∅)
2723, 26sylbi 220 . . . . . . . . . 10 (𝑎 ∈ (ran 𝑡 ∖ {∅}) → ran 𝑡 ≠ ∅)
2827exlimiv 1931 . . . . . . . . 9 (∃𝑎 𝑎 ∈ (ran 𝑡 ∖ {∅}) → ran 𝑡 ≠ ∅)
2922, 28sylbi 220 . . . . . . . 8 ((ran 𝑡 ∖ {∅}) ≠ ∅ → ran 𝑡 ≠ ∅)
3021, 29syl 17 . . . . . . 7 (𝑡:ω–1-1𝑉 ran 𝑡 ≠ ∅)
311fin23lem14 9748 . . . . . . 7 ((𝑎 ∈ ω ∧ ran 𝑡 ≠ ∅) → (𝑈𝑎) ≠ ∅)
327, 30, 31syl2anr 599 . . . . . 6 ((𝑡:ω–1-1𝑉𝑎 ∈ ω) → (𝑈𝑎) ≠ ∅)
33 neeq1 3052 . . . . . 6 ((𝑈𝑎) = ran 𝑈 → ((𝑈𝑎) ≠ ∅ ↔ ran 𝑈 ≠ ∅))
3432, 33syl5ibcom 248 . . . . 5 ((𝑡:ω–1-1𝑉𝑎 ∈ ω) → ((𝑈𝑎) = ran 𝑈 ran 𝑈 ≠ ∅))
3534rexlimdva 3246 . . . 4 (𝑡:ω–1-1𝑉 → (∃𝑎 ∈ ω (𝑈𝑎) = ran 𝑈 ran 𝑈 ≠ ∅))
366, 35syl5bi 245 . . 3 (𝑡:ω–1-1𝑉 → ( ran 𝑈 ∈ ran 𝑈 ran 𝑈 ≠ ∅))
3736adantl 485 . 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 209  wa 399   = wceq 1538  wex 1781  wcel 2112  {cab 2779  wne 2990  wral 3109  wrex 3110  Vcvv 3444  cdif 3881  cin 3883  wss 3884  c0 4246  ifcif 4428  𝒫 cpw 4500  {csn 4528   cuni 4803   cint 4841   class class class wbr 5033  ran crn 5524  suc csuc 6165   Fn wfn 6323  1-1wf1 6325  1-1-ontowf1o 6327  cfv 6328  (class class class)co 7139  cmpo 7141  ωcom 7564  seqωcseqom 8070  m cmap 8393  cen 8493  Fincfn 8496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-seqom 8071  df-1o 8089  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500
This theorem is referenced by:  fin23lem31  9758
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