Theorem isf33lem 10288

Description: Lemma for isfin3-3 10290. (Contributed by Stefan O'Rear, 17-May-2015.)

Assertion

Ref	Expression
isf33lem	⊢ FinIII = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}

Distinct variable group:   𝑔,𝑎,𝑥

Proof of Theorem isf33lem

Dummy variables 𝑏 𝑓 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isfin32i 10287	. . . 4 ⊢ (𝑓 ∈ FinIII → ¬ ω ≼* 𝑓)
2		fveq1 6839	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → (𝑎‘suc 𝑥) = (𝑏‘suc 𝑥))
3		fveq1 6839	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → (𝑎‘𝑥) = (𝑏‘𝑥))
4	2, 3	sseq12d 3955	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → ((𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) ↔ (𝑏‘suc 𝑥) ⊆ (𝑏‘𝑥)))
5	4	ralbidv 3160	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) ↔ ∀𝑥 ∈ ω (𝑏‘suc 𝑥) ⊆ (𝑏‘𝑥)))
6		rneq 5891	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → ran 𝑎 = ran 𝑏)
7	6	inteqd 4894	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → ∩ ran 𝑎 = ∩ ran 𝑏)
8	7, 6	eleq12d 2830	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (∩ ran 𝑎 ∈ ran 𝑎 ↔ ∩ ran 𝑏 ∈ ran 𝑏))
9	5, 8	imbi12d 344	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → ((∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎) ↔ (∀𝑥 ∈ ω (𝑏‘suc 𝑥) ⊆ (𝑏‘𝑥) → ∩ ran 𝑏 ∈ ran 𝑏)))
10	9	cbvralvw 3215	. . . . . . 7 ⊢ (∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎) ↔ ∀𝑏 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑏‘suc 𝑥) ⊆ (𝑏‘𝑥) → ∩ ran 𝑏 ∈ ran 𝑏))
11		pweq 4555	. . . . . . . . 9 ⊢ (𝑔 = 𝑦 → 𝒫 𝑔 = 𝒫 𝑦)
12	11	oveq1d 7382	. . . . . . . 8 ⊢ (𝑔 = 𝑦 → (𝒫 𝑔 ↑m ω) = (𝒫 𝑦 ↑m ω))
13	12	raleqdv 3295	. . . . . . 7 ⊢ (𝑔 = 𝑦 → (∀𝑏 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑏‘suc 𝑥) ⊆ (𝑏‘𝑥) → ∩ ran 𝑏 ∈ ran 𝑏) ↔ ∀𝑏 ∈ (𝒫 𝑦 ↑m ω)(∀𝑥 ∈ ω (𝑏‘suc 𝑥) ⊆ (𝑏‘𝑥) → ∩ ran 𝑏 ∈ ran 𝑏)))
14	10, 13	bitrid 283	. . . . . 6 ⊢ (𝑔 = 𝑦 → (∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎) ↔ ∀𝑏 ∈ (𝒫 𝑦 ↑m ω)(∀𝑥 ∈ ω (𝑏‘suc 𝑥) ⊆ (𝑏‘𝑥) → ∩ ran 𝑏 ∈ ran 𝑏)))
15	14	cbvabv 2806	. . . . 5 ⊢ {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)} = {𝑦 ∣ ∀𝑏 ∈ (𝒫 𝑦 ↑m ω)(∀𝑥 ∈ ω (𝑏‘suc 𝑥) ⊆ (𝑏‘𝑥) → ∩ ran 𝑏 ∈ ran 𝑏)}
16	15	isf32lem12 10286	. . . 4 ⊢ (𝑓 ∈ FinIII → (¬ ω ≼* 𝑓 → 𝑓 ∈ {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}))
17	1, 16	mpd 15	. . 3 ⊢ (𝑓 ∈ FinIII → 𝑓 ∈ {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)})
18	10	abbii 2803	. . . 4 ⊢ {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)} = {𝑔 ∣ ∀𝑏 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑏‘suc 𝑥) ⊆ (𝑏‘𝑥) → ∩ ran 𝑏 ∈ ran 𝑏)}
19	18	fin23lem41 10274	. . 3 ⊢ (𝑓 ∈ {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)} → 𝑓 ∈ FinIII)
20	17, 19	impbii 209	. 2 ⊢ (𝑓 ∈ FinIII ↔ 𝑓 ∈ {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)})
21	20	eqriv 2733	1 ⊢ FinIII = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   ∈ wcel 2114  {cab 2714  ∀wral 3051   ⊆ wss 3889  𝒫 cpw 4541  ∩ cint 4889   class class class wbr 5085  ran crn 5632  suc csuc 6325  ‘cfv 6498  (class class class)co 7367  ωcom 7817   ↑_m cmap 8773   ≼^* cwdom 9479  Fin^IIIcfin3 10203

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  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 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-seqom 8387  df-1o 8405  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-wdom 9480  df-card 9863  df-fin4 10209  df-fin3 10210

This theorem is referenced by:  isfin3-2  10289  isfin3-3  10290  fin23  10311