Theorem fin23lem33 9167

Description: Lemma for fin23 9211. 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 6191	. . . . . . 7 ⊢ (𝑗 = 𝑐 → (𝑒‘𝑗) = (𝑒‘𝑐))
2	1	ineq1d 3813	. . . . . 6 ⊢ (𝑗 = 𝑐 → ((𝑒‘𝑗) ∩ 𝑘) = ((𝑒‘𝑐) ∩ 𝑘))
3	2	eqeq1d 2624	. . . . 5 ⊢ (𝑗 = 𝑐 → (((𝑒‘𝑗) ∩ 𝑘) = ∅ ↔ ((𝑒‘𝑐) ∩ 𝑘) = ∅))
4	3, 2	ifbieq2d 4111	. . . 4 ⊢ (𝑗 = 𝑐 → if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘)) = if(((𝑒‘𝑐) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑐) ∩ 𝑘)))
5		ineq2 3808	. . . . . 6 ⊢ (𝑘 = 𝑑 → ((𝑒‘𝑐) ∩ 𝑘) = ((𝑒‘𝑐) ∩ 𝑑))
6	5	eqeq1d 2624	. . . . 5 ⊢ (𝑘 = 𝑑 → (((𝑒‘𝑐) ∩ 𝑘) = ∅ ↔ ((𝑒‘𝑐) ∩ 𝑑) = ∅))
7		id 22	. . . . 5 ⊢ (𝑘 = 𝑑 → 𝑘 = 𝑑)
8	6, 7, 5	ifbieq12d 4113	. . . 4 ⊢ (𝑘 = 𝑑 → if(((𝑒‘𝑐) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑐) ∩ 𝑘)) = if(((𝑒‘𝑐) ∩ 𝑑) = ∅, 𝑑, ((𝑒‘𝑐) ∩ 𝑑)))
9	4, 8	cbvmpt2v 6735	. . 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 2622	. . 3 ⊢ ∪ ran 𝑒 = ∪ ran 𝑒
11		seqomeq12 7549	. . 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 708	. 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 6191	. . . 4 ⊢ (𝑙 = 𝑦 → (𝑒‘𝑙) = (𝑒‘𝑦))
15	14	sseq2d 3633	. . 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 3199	. 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 2622	. 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 2622	. 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 2622	. 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 9166	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 384  ∀wal 1481   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608  ∀wral 2912  {crab 2916  Vcvv 3200   ∖ cdif 3571   ∩ cin 3573   ⊆ wss 3574   ⊊ wpss 3575  ∅c0 3915  ifcif 4086  𝒫 cpw 4158  ∪ cuni 4436  ∩ cint 4475   class class class wbr 4653   ↦ cmpt 4729  ran crn 5115   ∘ ccom 5118  suc csuc 5725  –1-1→wf1 5885  ‘cfv 5888  ℩crio 6610  (class class class)co 6650   ↦ cmpt2 6652  ωcom 7065  seq_𝜔cseqom 7542   ↑_𝑚 cmap 7857   ≈ cen 7952  Fincfn 7955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765

This theorem is referenced by:  fin23lem41  9174