Theorem isf32lem12 9586

Description: Lemma for isfin3-2 9589. (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 8230	. . . . 5 ⊢ (𝑓 ∈ (𝒫 𝐺 ↑𝑚 ω) → 𝑓:ω⟶𝒫 𝐺)
2		isf32lem11 9585	. . . . . . . . . 10 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑓:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) ∧ ¬ ∩ ran 𝑓 ∈ ran 𝑓)) → ω ≼* 𝐺)
3	2	expcom 406	. . . . . . . . 9 ⊢ ((𝑓:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) ∧ ¬ ∩ ran 𝑓 ∈ ran 𝑓) → (𝐺 ∈ 𝑉 → ω ≼* 𝐺))
4	3	3expa 1098	. . . . . . . 8 ⊢ (((𝑓:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏)) ∧ ¬ ∩ ran 𝑓 ∈ ran 𝑓) → (𝐺 ∈ 𝑉 → ω ≼* 𝐺))
5	4	impancom 444	. . . . . . 7 ⊢ (((𝑓:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏)) ∧ 𝐺 ∈ 𝑉) → (¬ ∩ ran 𝑓 ∈ ran 𝑓 → ω ≼* 𝐺))
6	5	con1d 142	. . . . . 6 ⊢ (((𝑓:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏)) ∧ 𝐺 ∈ 𝑉) → (¬ ω ≼* 𝐺 → ∩ ran 𝑓 ∈ ran 𝑓))
7	6	exp31 412	. . . . 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 3138	. 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 9551	. 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 251	1 ⊢ (𝐺 ∈ 𝑉 → (¬ ω ≼* 𝐺 → 𝐺 ∈ 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 387   ∧ w3a 1068   = wceq 1507   ∈ wcel 2050  {cab 2758  ∀wral 3088   ⊆ wss 3831  𝒫 cpw 4423  ∩ cint 4750   class class class wbr 4930  ran crn 5409  suc csuc 6033  ⟶wf 6186  ‘cfv 6190  (class class class)co 6978  ωcom 7398   ↑_𝑚 cmap 8208   ≼^* cwdom 8818

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5050  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-pss 3847  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-tp 4447  df-op 4449  df-uni 4714  df-int 4751  df-iun 4795  df-br 4931  df-opab 4993  df-mpt 5010  df-tr 5032  df-id 5313  df-eprel 5318  df-po 5327  df-so 5328  df-fr 5367  df-se 5368  df-we 5369  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-pred 5988  df-ord 6034  df-on 6035  df-lim 6036  df-suc 6037  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-f1 6195  df-fo 6196  df-f1o 6197  df-fv 6198  df-isom 6199  df-riota 6939  df-ov 6981  df-oprab 6982  df-mpo 6983  df-om 7399  df-1st 7503  df-2nd 7504  df-wrecs 7752  df-recs 7814  df-1o 7907  df-er 8091  df-map 8210  df-en 8309  df-dom 8310  df-sdom 8311  df-fin 8312  df-wdom 8820  df-card 9164

This theorem is referenced by:  isf33lem  9588  isfin3-2  9589