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Theorem isf32lem12 9267
Description: Lemma for isfin3-2 9270. (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 7964 . . . . 5 (𝑓 ∈ (𝒫 𝐺𝑚 ω) → 𝑓:ω⟶𝒫 𝐺)
2 isf32lem11 9266 . . . . . . . . . 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 1111 . . . . . . . 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 631 . . . . 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 3038 . 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 9232 . 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 1072   = wceq 1564  wcel 2071  {cab 2678  wral 2982  wss 3648  𝒫 cpw 4234   cint 4551   class class class wbr 4728  ran crn 5187  suc csuc 5806  wf 5965  cfv 5969  (class class class)co 6733  ωcom 7150  𝑚 cmap 7942  * cwdom 8546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1818  ax-5 1920  ax-6 1986  ax-7 2022  ax-8 2073  ax-9 2080  ax-10 2100  ax-11 2115  ax-12 2128  ax-13 2323  ax-ext 2672  ax-rep 4847  ax-sep 4857  ax-nul 4865  ax-pow 4916  ax-pr 4979  ax-un 7034
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1567  df-ex 1786  df-nf 1791  df-sb 1979  df-eu 2543  df-mo 2544  df-clab 2679  df-cleq 2685  df-clel 2688  df-nfc 2823  df-ne 2865  df-ral 2987  df-rex 2988  df-reu 2989  df-rmo 2990  df-rab 2991  df-v 3274  df-sbc 3510  df-csb 3608  df-dif 3651  df-un 3653  df-in 3655  df-ss 3662  df-pss 3664  df-nul 3992  df-if 4163  df-pw 4236  df-sn 4254  df-pr 4256  df-tp 4258  df-op 4260  df-uni 4513  df-int 4552  df-iun 4598  df-br 4729  df-opab 4789  df-mpt 4806  df-tr 4829  df-id 5096  df-eprel 5101  df-po 5107  df-so 5108  df-fr 5145  df-se 5146  df-we 5147  df-xp 5192  df-rel 5193  df-cnv 5194  df-co 5195  df-dm 5196  df-rn 5197  df-res 5198  df-ima 5199  df-pred 5761  df-ord 5807  df-on 5808  df-lim 5809  df-suc 5810  df-iota 5932  df-fun 5971  df-fn 5972  df-f 5973  df-f1 5974  df-fo 5975  df-f1o 5976  df-fv 5977  df-isom 5978  df-riota 6694  df-ov 6736  df-oprab 6737  df-mpt2 6738  df-om 7151  df-1st 7253  df-2nd 7254  df-wrecs 7495  df-recs 7556  df-1o 7648  df-er 7830  df-map 7944  df-en 8041  df-dom 8042  df-sdom 8043  df-fin 8044  df-wdom 8548  df-card 8846
This theorem is referenced by:  isf33lem  9269  isfin3-2  9270
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