Theorem isf32lem12 10051

Description: Lemma for isfin3-2 10054. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)

Hypothesis

Ref	Expression
isf32lem40.f	⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}

Assertion

Ref	Expression
isf32lem12	⊢ (𝐺 ∈ 𝑉 → (¬ ω ≼* 𝐺 → 𝐺 ∈ 𝐹))

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.

Step	Hyp	Ref	Expression
1		elmapi 8595	. . . . 5 ⊢ (𝑓 ∈ (𝒫 𝐺 ↑m ω) → 𝑓:ω⟶𝒫 𝐺)
2		isf32lem11 10050	. . . . . . . . . 10 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑓:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) ∧ ¬ ∩ ran 𝑓 ∈ ran 𝑓)) → ω ≼* 𝐺)
3	2	expcom 413	. . . . . . . . 9 ⊢ ((𝑓:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) ∧ ¬ ∩ ran 𝑓 ∈ ran 𝑓) → (𝐺 ∈ 𝑉 → ω ≼* 𝐺))
4	3	3expa 1116	. . . . . . . 8 ⊢ (((𝑓:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏)) ∧ ¬ ∩ ran 𝑓 ∈ ran 𝑓) → (𝐺 ∈ 𝑉 → ω ≼* 𝐺))
5	4	impancom 451	. . . . . . 7 ⊢ (((𝑓:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏)) ∧ 𝐺 ∈ 𝑉) → (¬ ∩ ran 𝑓 ∈ ran 𝑓 → ω ≼* 𝐺))
6	5	con1d 145	. . . . . 6 ⊢ (((𝑓:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏)) ∧ 𝐺 ∈ 𝑉) → (¬ ω ≼* 𝐺 → ∩ ran 𝑓 ∈ ran 𝑓))
7	6	exp31 419	. . . . 5 ⊢ (𝑓:ω⟶𝒫 𝐺 → (∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) → (𝐺 ∈ 𝑉 → (¬ ω ≼* 𝐺 → ∩ ran 𝑓 ∈ ran 𝑓))))
8	1, 7	syl 17	. . . 4 ⊢ (𝑓 ∈ (𝒫 𝐺 ↑m ω) → (∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) → (𝐺 ∈ 𝑉 → (¬ ω ≼* 𝐺 → ∩ ran 𝑓 ∈ ran 𝑓))))
9	8	com4t 93	. . 3 ⊢ (𝐺 ∈ 𝑉 → (¬ ω ≼* 𝐺 → (𝑓 ∈ (𝒫 𝐺 ↑m ω) → (∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) → ∩ ran 𝑓 ∈ ran 𝑓))))
10	9	ralrimdv 3111	. 2 ⊢ (𝐺 ∈ 𝑉 → (¬ ω ≼* 𝐺 → ∀𝑓 ∈ (𝒫 𝐺 ↑m ω)(∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) → ∩ ran 𝑓 ∈ ran 𝑓)))
11		isf32lem40.f	. . 3 ⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}
12	11	isfin3ds 10016	. 2 ⊢ (𝐺 ∈ 𝑉 → (𝐺 ∈ 𝐹 ↔ ∀𝑓 ∈ (𝒫 𝐺 ↑m ω)(∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) → ∩ ran 𝑓 ∈ ran 𝑓)))
13	10, 12	sylibrd 258	1 ⊢ (𝐺 ∈ 𝑉 → (¬ ω ≼* 𝐺 → 𝐺 ∈ 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  {cab 2715  ∀wral 3063   ⊆ wss 3883  𝒫 cpw 4530  ∩ cint 4876   class class class wbr 5070  ran crn 5581  suc csuc 6253  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  ωcom 7687   ↑_m cmap 8573   ≼^* cwdom 9253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-1o 8267  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-wdom 9254  df-card 9628

This theorem is referenced by:  isf33lem  10053  isfin3-2  10054