Theorem fin23lem33 9502

Description: Lemma for fin23 9546. 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 6446	. . . . . . 7 ⊢ (𝑗 = 𝑐 → (𝑒‘𝑗) = (𝑒‘𝑐))
2	1	ineq1d 4036	. . . . . 6 ⊢ (𝑗 = 𝑐 → ((𝑒‘𝑗) ∩ 𝑘) = ((𝑒‘𝑐) ∩ 𝑘))
3	2	eqeq1d 2780	. . . . 5 ⊢ (𝑗 = 𝑐 → (((𝑒‘𝑗) ∩ 𝑘) = ∅ ↔ ((𝑒‘𝑐) ∩ 𝑘) = ∅))
4	3, 2	ifbieq2d 4332	. . . 4 ⊢ (𝑗 = 𝑐 → if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘)) = if(((𝑒‘𝑐) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑐) ∩ 𝑘)))
5		ineq2 4031	. . . . . 6 ⊢ (𝑘 = 𝑑 → ((𝑒‘𝑐) ∩ 𝑘) = ((𝑒‘𝑐) ∩ 𝑑))
6	5	eqeq1d 2780	. . . . 5 ⊢ (𝑘 = 𝑑 → (((𝑒‘𝑐) ∩ 𝑘) = ∅ ↔ ((𝑒‘𝑐) ∩ 𝑑) = ∅))
7		id 22	. . . . 5 ⊢ (𝑘 = 𝑑 → 𝑘 = 𝑑)
8	6, 7, 5	ifbieq12d 4334	. . . 4 ⊢ (𝑘 = 𝑑 → if(((𝑒‘𝑐) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑐) ∩ 𝑘)) = if(((𝑒‘𝑐) ∩ 𝑑) = ∅, 𝑑, ((𝑒‘𝑐) ∩ 𝑑)))
9	4, 8	cbvmpt2v 7012	. . 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 2778	. . 3 ⊢ ∪ ran 𝑒 = ∪ ran 𝑒
11		seqomeq12 7832	. . 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 682	. 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 6446	. . . 4 ⊢ (𝑙 = 𝑦 → (𝑒‘𝑙) = (𝑒‘𝑦))
15	14	sseq2d 3852	. . 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 3396	. 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 2778	. 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 2778	. 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 2778	. 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 9501	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 386  ∀wal 1599   = wceq 1601  ∃wex 1823   ∈ wcel 2107  {cab 2763  ∀wral 3090  {crab 3094  Vcvv 3398   ∖ cdif 3789   ∩ cin 3791   ⊆ wss 3792   ⊊ wpss 3793  ∅c0 4141  ifcif 4307  𝒫 cpw 4379  ∪ cuni 4671  ∩ cint 4710   class class class wbr 4886   ↦ cmpt 4965  ran crn 5356   ∘ ccom 5359  suc csuc 5978  –1-1→wf1 6132  ‘cfv 6135  ℩crio 6882  (class class class)co 6922   ↦ cmpt2 6924  ωcom 7343  seq_𝜔cseqom 7825   ↑_𝑚 cmap 8140   ≈ cen 8238  Fincfn 8241

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-seqom 7826  df-1o 7843  df-oadd 7847  df-er 8026  df-map 8142  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-card 9098

This theorem is referenced by:  fin23lem41  9509