Theorem fin23lem33 10392

Description: Lemma for fin23 10436. Discharge hypotheses. (Contributed by Stefan O'Rear, 2-Nov-2014.)

Hypothesis

Ref	Expression
fin23lem33.f	⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}

Assertion

Ref	Expression
fin23lem33	⊢ (𝐺 ∈ 𝐹 → ∃𝑓∀𝑏((𝑏:ω–1-1→V ∧ ∪ ran 𝑏 ⊆ 𝐺) → ((𝑓‘𝑏):ω–1-1→V ∧ ∪ ran (𝑓‘𝑏) ⊊ ∪ ran 𝑏)))

Distinct variable groups:   𝑎,𝑏,𝑓,𝑔,𝑥,𝐺   𝐹,𝑎

Allowed substitution hints:   𝐹(𝑥,𝑓,𝑔,𝑏)

Proof of Theorem fin23lem33

Dummy variables 𝑐 𝑑 𝑒 𝑖 𝑗 𝑘 𝑙 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6914	. . . . . . 7 ⊢ (𝑗 = 𝑐 → (𝑒‘𝑗) = (𝑒‘𝑐))
2	1	ineq1d 4230	. . . . . 6 ⊢ (𝑗 = 𝑐 → ((𝑒‘𝑗) ∩ 𝑘) = ((𝑒‘𝑐) ∩ 𝑘))
3	2	eqeq1d 2739	. . . . 5 ⊢ (𝑗 = 𝑐 → (((𝑒‘𝑗) ∩ 𝑘) = ∅ ↔ ((𝑒‘𝑐) ∩ 𝑘) = ∅))
4	3, 2	ifbieq2d 4560	. . . 4 ⊢ (𝑗 = 𝑐 → if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘)) = if(((𝑒‘𝑐) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑐) ∩ 𝑘)))
5		ineq2 4225	. . . . . 6 ⊢ (𝑘 = 𝑑 → ((𝑒‘𝑐) ∩ 𝑘) = ((𝑒‘𝑐) ∩ 𝑑))
6	5	eqeq1d 2739	. . . . 5 ⊢ (𝑘 = 𝑑 → (((𝑒‘𝑐) ∩ 𝑘) = ∅ ↔ ((𝑒‘𝑐) ∩ 𝑑) = ∅))
7		id 22	. . . . 5 ⊢ (𝑘 = 𝑑 → 𝑘 = 𝑑)
8	6, 7, 5	ifbieq12d 4562	. . . 4 ⊢ (𝑘 = 𝑑 → if(((𝑒‘𝑐) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑐) ∩ 𝑘)) = if(((𝑒‘𝑐) ∩ 𝑑) = ∅, 𝑑, ((𝑒‘𝑐) ∩ 𝑑)))
9	4, 8	cbvmpov 7535	. . 3 ⊢ (𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))) = (𝑐 ∈ ω, 𝑑 ∈ V ↦ if(((𝑒‘𝑐) ∩ 𝑑) = ∅, 𝑑, ((𝑒‘𝑐) ∩ 𝑑)))
10		eqid 2737	. . 3 ⊢ ∪ ran 𝑒 = ∪ ran 𝑒
11		seqomeq12 8502	. . 3 ⊢ (((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))) = (𝑐 ∈ ω, 𝑑 ∈ V ↦ if(((𝑒‘𝑐) ∩ 𝑑) = ∅, 𝑑, ((𝑒‘𝑐) ∩ 𝑑))) ∧ ∪ ran 𝑒 = ∪ ran 𝑒) → seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) = seqω((𝑐 ∈ ω, 𝑑 ∈ V ↦ if(((𝑒‘𝑐) ∩ 𝑑) = ∅, 𝑑, ((𝑒‘𝑐) ∩ 𝑑))), ∪ ran 𝑒))
12	9, 10, 11	mp2an 692	. 2 ⊢ seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) = seqω((𝑐 ∈ ω, 𝑑 ∈ V ↦ if(((𝑒‘𝑐) ∩ 𝑑) = ∅, 𝑑, ((𝑒‘𝑐) ∩ 𝑑))), ∪ ran 𝑒)
13		fin23lem33.f	. 2 ⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}
14		fveq2 6914	. . . 4 ⊢ (𝑙 = 𝑦 → (𝑒‘𝑙) = (𝑒‘𝑦))
15	14	sseq2d 4031	. . 3 ⊢ (𝑙 = 𝑦 → (∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙) ↔ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑦)))
16	15	cbvrabv 3447	. 2 ⊢ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)} = {𝑦 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑦)}
17		eqid 2737	. 2 ⊢ (𝑔 ∈ ω ↦ (℩𝑥 ∈ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)} (𝑥 ∩ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)}) ≈ 𝑔)) = (𝑔 ∈ ω ↦ (℩𝑥 ∈ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)} (𝑥 ∩ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)}) ≈ 𝑔))
18		eqid 2737	. 2 ⊢ (𝑔 ∈ ω ↦ (℩𝑥 ∈ (ω ∖ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)})(𝑥 ∩ (ω ∖ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)})) ≈ 𝑔)) = (𝑔 ∈ ω ↦ (℩𝑥 ∈ (ω ∖ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)})(𝑥 ∩ (ω ∖ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)})) ≈ 𝑔))
19		eqid 2737	. 2 ⊢ if({𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)} ∈ Fin, (𝑒 ∘ (𝑔 ∈ ω ↦ (℩𝑥 ∈ (ω ∖ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)})(𝑥 ∩ (ω ∖ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)})) ≈ 𝑔))), ((𝑖 ∈ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)} ↦ ((𝑒‘𝑖) ∖ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒))) ∘ (𝑔 ∈ ω ↦ (℩𝑥 ∈ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)} (𝑥 ∩ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)}) ≈ 𝑔)))) = if({𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)} ∈ Fin, (𝑒 ∘ (𝑔 ∈ ω ↦ (℩𝑥 ∈ (ω ∖ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)})(𝑥 ∩ (ω ∖ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)})) ≈ 𝑔))), ((𝑖 ∈ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)} ↦ ((𝑒‘𝑖) ∖ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒))) ∘ (𝑔 ∈ ω ↦ (℩𝑥 ∈ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)} (𝑥 ∩ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)}) ≈ 𝑔))))
20	12, 13, 16, 17, 18, 19	fin23lem32 10391	1 ⊢ (𝐺 ∈ 𝐹 → ∃𝑓∀𝑏((𝑏:ω–1-1→V ∧ ∪ ran 𝑏 ⊆ 𝐺) → ((𝑓‘𝑏):ω–1-1→V ∧ ∪ ran (𝑓‘𝑏) ⊊ ∪ ran 𝑏)))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1537   = wceq 1539  ∃wex 1778   ∈ wcel 2108  {cab 2714  ∀wral 3061  {crab 3436  Vcvv 3481   ∖ cdif 3963   ∩ cin 3965   ⊆ wss 3966   ⊊ wpss 3967  ∅c0 4342  ifcif 4534  𝒫 cpw 4608  ∪ cuni 4915  ∩ cint 4954   class class class wbr 5151   ↦ cmpt 5234  ran crn 5694   ∘ ccom 5697  suc csuc 6394  –1-1→wf1 6566  ‘cfv 6569  ℩crio 7394  (class class class)co 7438   ∈ cmpo 7440  ωcom 7894  seq_ωcseqom 8495   ↑_m cmap 8874   ≈ cen 8990  Fincfn 8993

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  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 2708  ax-rep 5288  ax-sep 5305  ax-nul 5315  ax-pow 5374  ax-pr 5441  ax-un 7761

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3483  df-sbc 3795  df-csb 3912  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-pss 3986  df-nul 4343  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-int 4955  df-iun 5001  df-br 5152  df-opab 5214  df-mpt 5235  df-tr 5269  df-id 5587  df-eprel 5593  df-po 5601  df-so 5602  df-fr 5645  df-se 5646  df-we 5647  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-res 5705  df-ima 5706  df-pred 6329  df-ord 6395  df-on 6396  df-lim 6397  df-suc 6398  df-iota 6522  df-fun 6571  df-fn 6572  df-f 6573  df-f1 6574  df-fo 6575  df-f1o 6576  df-fv 6577  df-isom 6578  df-riota 7395  df-ov 7441  df-oprab 7442  df-mpo 7443  df-om 7895  df-1st 8022  df-2nd 8023  df-frecs 8314  df-wrecs 8345  df-recs 8419  df-rdg 8458  df-seqom 8496  df-1o 8514  df-er 8753  df-map 8876  df-en 8994  df-dom 8995  df-sdom 8996  df-fin 8997  df-card 9986

This theorem is referenced by:  fin23lem41  10399