MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isf32lem12 Structured version   Visualization version   GIF version

Theorem isf32lem12 9131
Description: Lemma for isfin3-2 9134. (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 7824 . . . . 5 (𝑓 ∈ (𝒫 𝐺𝑚 ω) → 𝑓:ω⟶𝒫 𝐺)
2 isf32lem11 9130 . . . . . . . . . 10 ((𝐺𝑉 ∧ (𝑓:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏) ∧ ¬ ran 𝑓 ∈ ran 𝑓)) → ω ≼* 𝐺)
32expcom 451 . . . . . . . . 9 ((𝑓:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏) ∧ ¬ ran 𝑓 ∈ ran 𝑓) → (𝐺𝑉 → ω ≼* 𝐺))
433expa 1262 . . . . . . . 8 (((𝑓:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏)) ∧ ¬ ran 𝑓 ∈ ran 𝑓) → (𝐺𝑉 → ω ≼* 𝐺))
54impancom 456 . . . . . . 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 2967 . 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 9096 . 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 384  w3a 1036   = wceq 1480  wcel 1992  {cab 2612  wral 2912  wss 3560  𝒫 cpw 4135   cint 4445   class class class wbr 4618  ran crn 5080  suc csuc 5687  wf 5846  cfv 5850  (class class class)co 6605  ωcom 7013  𝑚 cmap 7803  * cwdom 8407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-1o 7506  df-er 7688  df-map 7805  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-wdom 8409  df-card 8710
This theorem is referenced by:  isf33lem  9133  isfin3-2  9134
  Copyright terms: Public domain W3C validator