Theorem isf33lem 10326

Description: Lemma for isfin3-3 10328. (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 10325	. . . 4 ⊢ (𝑓 ∈ FinIII → ¬ ω ≼* 𝑓)
2		fveq1 6860	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → (𝑎‘suc 𝑥) = (𝑏‘suc 𝑥))
3		fveq1 6860	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → (𝑎‘𝑥) = (𝑏‘𝑥))
4	2, 3	sseq12d 3983	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → ((𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) ↔ (𝑏‘suc 𝑥) ⊆ (𝑏‘𝑥)))
5	4	ralbidv 3157	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) ↔ ∀𝑥 ∈ ω (𝑏‘suc 𝑥) ⊆ (𝑏‘𝑥)))
6		rneq 5903	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → ran 𝑎 = ran 𝑏)
7	6	inteqd 4918	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → ∩ ran 𝑎 = ∩ ran 𝑏)
8	7, 6	eleq12d 2823	. . . . . . . . 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 3216	. . . . . . 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 4580	. . . . . . . . 9 ⊢ (𝑔 = 𝑦 → 𝒫 𝑔 = 𝒫 𝑦)
12	11	oveq1d 7405	. . . . . . . 8 ⊢ (𝑔 = 𝑦 → (𝒫 𝑔 ↑m ω) = (𝒫 𝑦 ↑m ω))
13	12	raleqdv 3301	. . . . . . 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 2800	. . . . 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 10324	. . . 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 2797	. . . 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 10312	. . 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 2727	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 2109  {cab 2708  ∀wral 3045   ⊆ wss 3917  𝒫 cpw 4566  ∩ cint 4913   class class class wbr 5110  ran crn 5642  suc csuc 6337  ‘cfv 6514  (class class class)co 7390  ωcom 7845   ↑_m cmap 8802   ≼^* cwdom 9524  Fin^IIIcfin3 10241

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-seqom 8419  df-1o 8437  df-er 8674  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-wdom 9525  df-card 9899  df-fin4 10247  df-fin3 10248

This theorem is referenced by:  isfin3-2  10327  isfin3-3  10328  fin23  10349