Theorem isf32lem12 9186

Description: Lemma for isfin3-2 9189. (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 7879	. . . . 5 ⊢ (𝑓 ∈ (𝒫 𝐺 ↑𝑚 ω) → 𝑓:ω⟶𝒫 𝐺)
2		isf32lem11 9185	. . . . . . . . . 10 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑓:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) ∧ ¬ ∩ ran 𝑓 ∈ ran 𝑓)) → ω ≼* 𝐺)
3	2	expcom 451	. . . . . . . . 9 ⊢ ((𝑓:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) ∧ ¬ ∩ ran 𝑓 ∈ ran 𝑓) → (𝐺 ∈ 𝑉 → ω ≼* 𝐺))
4	3	3expa 1265	. . . . . . . 8 ⊢ (((𝑓:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏)) ∧ ¬ ∩ ran 𝑓 ∈ ran 𝑓) → (𝐺 ∈ 𝑉 → ω ≼* 𝐺))
5	4	impancom 456	. . . . . . 7 ⊢ (((𝑓:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏)) ∧ 𝐺 ∈ 𝑉) → (¬ ∩ ran 𝑓 ∈ ran 𝑓 → ω ≼* 𝐺))
6	5	con1d 139	. . . . . 6 ⊢ (((𝑓:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏)) ∧ 𝐺 ∈ 𝑉) → (¬ ω ≼* 𝐺 → ∩ ran 𝑓 ∈ ran 𝑓))
7	6	exp31 630	. . . . 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 2968	. 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 9151	. 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 249	1 ⊢ (𝐺 ∈ 𝑉 → (¬ ω ≼* 𝐺 → 𝐺 ∈ 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  {cab 2608  ∀wral 2912   ⊆ wss 3574  𝒫 cpw 4158  ∩ cint 4475   class class class wbr 4653  ran crn 5115  suc csuc 5725  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  ωcom 7065   ↑_𝑚 cmap 7857   ≼^* cwdom 8462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-1o 7560  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-wdom 8464  df-card 8765

This theorem is referenced by:  isf33lem  9188  isfin3-2  9189