Theorem isf33lem 10122

Description: Lemma for isfin3-3 10124. (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 10121	. . . 4 ⊢ (𝑓 ∈ FinIII → ¬ ω ≼* 𝑓)
2		fveq1 6773	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → (𝑎‘suc 𝑥) = (𝑏‘suc 𝑥))
3		fveq1 6773	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → (𝑎‘𝑥) = (𝑏‘𝑥))
4	2, 3	sseq12d 3954	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → ((𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) ↔ (𝑏‘suc 𝑥) ⊆ (𝑏‘𝑥)))
5	4	ralbidv 3112	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) ↔ ∀𝑥 ∈ ω (𝑏‘suc 𝑥) ⊆ (𝑏‘𝑥)))
6		rneq 5845	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → ran 𝑎 = ran 𝑏)
7	6	inteqd 4884	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → ∩ ran 𝑎 = ∩ ran 𝑏)
8	7, 6	eleq12d 2833	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (∩ ran 𝑎 ∈ ran 𝑎 ↔ ∩ ran 𝑏 ∈ ran 𝑏))
9	5, 8	imbi12d 345	. . . . . . . 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 3383	. . . . . . 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 4549	. . . . . . . . 9 ⊢ (𝑔 = 𝑦 → 𝒫 𝑔 = 𝒫 𝑦)
12	11	oveq1d 7290	. . . . . . . 8 ⊢ (𝑔 = 𝑦 → (𝒫 𝑔 ↑m ω) = (𝒫 𝑦 ↑m ω))
13	12	raleqdv 3348	. . . . . . 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 282	. . . . . 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 2811	. . . . 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 10120	. . . 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 2808	. . . 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 10108	. . 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 208	. 2 ⊢ (𝑓 ∈ FinIII ↔ 𝑓 ∈ {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)})
21	20	eqriv 2735	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 1539   ∈ wcel 2106  {cab 2715  ∀wral 3064   ⊆ wss 3887  𝒫 cpw 4533  ∩ cint 4879   class class class wbr 5074  ran crn 5590  suc csuc 6268  ‘cfv 6433  (class class class)co 7275  ωcom 7712   ↑_m cmap 8615   ≼^* cwdom 9323  Fin^IIIcfin3 10037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-seqom 8279  df-1o 8297  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-wdom 9324  df-card 9697  df-fin4 10043  df-fin3 10044

This theorem is referenced by:  isfin3-2  10123  isfin3-3  10124  fin23  10145