Theorem isf33lem 10380

Description: Lemma for isfin3-3 10382. (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 10379	. . . 4 ⊢ (𝑓 ∈ FinIII → ¬ ω ≼* 𝑓)
2		fveq1 6875	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → (𝑎‘suc 𝑥) = (𝑏‘suc 𝑥))
3		fveq1 6875	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → (𝑎‘𝑥) = (𝑏‘𝑥))
4	2, 3	sseq12d 3992	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → ((𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) ↔ (𝑏‘suc 𝑥) ⊆ (𝑏‘𝑥)))
5	4	ralbidv 3163	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) ↔ ∀𝑥 ∈ ω (𝑏‘suc 𝑥) ⊆ (𝑏‘𝑥)))
6		rneq 5916	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → ran 𝑎 = ran 𝑏)
7	6	inteqd 4927	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → ∩ ran 𝑎 = ∩ ran 𝑏)
8	7, 6	eleq12d 2828	. . . . . . . . 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 3220	. . . . . . 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 4589	. . . . . . . . 9 ⊢ (𝑔 = 𝑦 → 𝒫 𝑔 = 𝒫 𝑦)
12	11	oveq1d 7420	. . . . . . . 8 ⊢ (𝑔 = 𝑦 → (𝒫 𝑔 ↑m ω) = (𝒫 𝑦 ↑m ω))
13	12	raleqdv 3305	. . . . . . 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 2805	. . . . 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 10378	. . . 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 2802	. . . 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 10366	. . 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 2732	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 1540   ∈ wcel 2108  {cab 2713  ∀wral 3051   ⊆ wss 3926  𝒫 cpw 4575  ∩ cint 4922   class class class wbr 5119  ran crn 5655  suc csuc 6354  ‘cfv 6531  (class class class)co 7405  ωcom 7861   ↑_m cmap 8840   ≼^* cwdom 9578  Fin^IIIcfin3 10295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-seqom 8462  df-1o 8480  df-er 8719  df-map 8842  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-wdom 9579  df-card 9953  df-fin4 10301  df-fin3 10302

This theorem is referenced by:  isfin3-2  10381  isfin3-3  10382  fin23  10403