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Theorem isf32lem12 9779
 Description: Lemma for isfin3-2 9782. (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 8415 . . . . 5 (𝑓 ∈ (𝒫 𝐺m ω) → 𝑓:ω⟶𝒫 𝐺)
2 isf32lem11 9778 . . . . . . . . . 10 ((𝐺𝑉 ∧ (𝑓:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏) ∧ ¬ ran 𝑓 ∈ ran 𝑓)) → ω ≼* 𝐺)
32expcom 417 . . . . . . . . 9 ((𝑓:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏) ∧ ¬ ran 𝑓 ∈ ran 𝑓) → (𝐺𝑉 → ω ≼* 𝐺))
433expa 1115 . . . . . . . 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 147 . . . . . 6 (((𝑓:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏)) ∧ 𝐺𝑉) → (¬ ω ≼* 𝐺 ran 𝑓 ∈ ran 𝑓))
76exp31 423 . . . . 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 3156 . 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 9744 . 2 (𝐺𝑉 → (𝐺𝐹 ↔ ∀𝑓 ∈ (𝒫 𝐺m ω)(∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏) → ran 𝑓 ∈ ran 𝑓)))
1310, 12sylibrd 262 1 (𝐺𝑉 → (¬ ω ≼* 𝐺𝐺𝐹))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112  {cab 2779  ∀wral 3109   ⊆ wss 3884  𝒫 cpw 4500  ∩ cint 4841   class class class wbr 5033  ran crn 5524  suc csuc 6165  ⟶wf 6324  ‘cfv 6328  (class class class)co 7139  ωcom 7564   ↑m cmap 8393   ≼* cwdom 9016 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-rep 5157  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-rmo 3117  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-se 5483  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-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-1o 8089  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-wdom 9017  df-card 9356 This theorem is referenced by:  isf33lem  9781  isfin3-2  9782
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