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Theorem isf32lem12 9224
Description: Lemma for isfin3-2 9227. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
Hypothesis
Ref Expression
isf32lem40.f 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔𝑚 ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
Assertion
Ref Expression
isf32lem12 (𝐺𝑉 → (¬ ω ≼* 𝐺𝐺𝐹))
Distinct variable groups:   𝑔,𝐹   𝑔,𝑎,𝑥,𝐺
Allowed substitution hints:   𝐹(𝑥,𝑎)   𝑉(𝑥,𝑔,𝑎)

Proof of Theorem isf32lem12
Dummy variables 𝑏 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elmapi 7921 . . . . 5 (𝑓 ∈ (𝒫 𝐺𝑚 ω) → 𝑓:ω⟶𝒫 𝐺)
2 isf32lem11 9223 . . . . . . . . . 10 ((𝐺𝑉 ∧ (𝑓:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏) ∧ ¬ ran 𝑓 ∈ ran 𝑓)) → ω ≼* 𝐺)
32expcom 450 . . . . . . . . 9 ((𝑓:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏) ∧ ¬ ran 𝑓 ∈ ran 𝑓) → (𝐺𝑉 → ω ≼* 𝐺))
433expa 1284 . . . . . . . 8 (((𝑓:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏)) ∧ ¬ ran 𝑓 ∈ ran 𝑓) → (𝐺𝑉 → ω ≼* 𝐺))
54impancom 455 . . . . . . 7 (((𝑓:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏)) ∧ 𝐺𝑉) → (¬ ran 𝑓 ∈ ran 𝑓 → ω ≼* 𝐺))
65con1d 139 . . . . . 6 (((𝑓:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏)) ∧ 𝐺𝑉) → (¬ ω ≼* 𝐺 ran 𝑓 ∈ ran 𝑓))
76exp31 629 . . . . 5 (𝑓:ω⟶𝒫 𝐺 → (∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏) → (𝐺𝑉 → (¬ ω ≼* 𝐺 ran 𝑓 ∈ ran 𝑓))))
81, 7syl 17 . . . 4 (𝑓 ∈ (𝒫 𝐺𝑚 ω) → (∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏) → (𝐺𝑉 → (¬ ω ≼* 𝐺 ran 𝑓 ∈ ran 𝑓))))
98com4t 93 . . 3 (𝐺𝑉 → (¬ ω ≼* 𝐺 → (𝑓 ∈ (𝒫 𝐺𝑚 ω) → (∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏) → ran 𝑓 ∈ ran 𝑓))))
109ralrimdv 2997 . 2 (𝐺𝑉 → (¬ ω ≼* 𝐺 → ∀𝑓 ∈ (𝒫 𝐺𝑚 ω)(∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏) → ran 𝑓 ∈ ran 𝑓)))
11 isf32lem40.f . . 3 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔𝑚 ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
1211isfin3ds 9189 . 2 (𝐺𝑉 → (𝐺𝐹 ↔ ∀𝑓 ∈ (𝒫 𝐺𝑚 ω)(∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏) → ran 𝑓 ∈ ran 𝑓)))
1310, 12sylibrd 249 1 (𝐺𝑉 → (¬ ω ≼* 𝐺𝐺𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  {cab 2637  wral 2941  wss 3607  𝒫 cpw 4191   cint 4507   class class class wbr 4685  ran crn 5144  suc csuc 5763  wf 5922  cfv 5926  (class class class)co 6690  ωcom 7107  𝑚 cmap 7899  * cwdom 8503
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-rep 4804  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-rmo 2949  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-se 5103  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-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-1o 7605  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-wdom 8505  df-card 8803
This theorem is referenced by:  isf33lem  9226  isfin3-2  9227
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