Theorem isf32lem12 10274

Description: Lemma for isfin3-2 10277. (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 8786	. . . . 5 ⊢ (𝑓 ∈ (𝒫 𝐺 ↑m ω) → 𝑓:ω⟶𝒫 𝐺)
2		isf32lem11 10273	. . . . . . . . . 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 1118	. . . . . . . 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 3134	. 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 10239	. 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 259	1 ⊢ (𝐺 ∈ 𝑉 → (¬ ω ≼* 𝐺 → 𝐺 ∈ 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  {cab 2714  ∀wral 3051   ⊆ wss 3901  𝒫 cpw 4554  ∩ cint 4902   class class class wbr 5098  ran crn 5625  suc csuc 6319  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358  ωcom 7808   ↑_m cmap 8763   ≼^* cwdom 9469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-1o 8397  df-er 8635  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-wdom 9470  df-card 9851

This theorem is referenced by:  isf33lem  10276  isfin3-2  10277