Theorem fin23lem33 10350

Description: Lemma for fin23 10394. 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 6882	. . . . . . 7 ⊢ (𝑗 = 𝑐 → (𝑒‘𝑗) = (𝑒‘𝑐))
2	1	ineq1d 4168	. . . . . 6 ⊢ (𝑗 = 𝑐 → ((𝑒‘𝑗) ∩ 𝑘) = ((𝑒‘𝑐) ∩ 𝑘))
3	2	eqeq1d 2764	. . . . 5 ⊢ (𝑗 = 𝑐 → (((𝑒‘𝑗) ∩ 𝑘) = ∅ ↔ ((𝑒‘𝑐) ∩ 𝑘) = ∅))
4	3, 2	ifbieq2d 4512	. . . 4 ⊢ (𝑗 = 𝑐 → if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘)) = if(((𝑒‘𝑐) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑐) ∩ 𝑘)))
5		ineq2 4163	. . . . . 6 ⊢ (𝑘 = 𝑑 → ((𝑒‘𝑐) ∩ 𝑘) = ((𝑒‘𝑐) ∩ 𝑑))
6	5	eqeq1d 2764	. . . . 5 ⊢ (𝑘 = 𝑑 → (((𝑒‘𝑐) ∩ 𝑘) = ∅ ↔ ((𝑒‘𝑐) ∩ 𝑑) = ∅))
7		id 23	. . . . 5 ⊢ (𝑘 = 𝑑 → 𝑘 = 𝑑)
8	6, 7, 5	ifbieq12d 4514	. . . 4 ⊢ (𝑘 = 𝑑 → if(((𝑒‘𝑐) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑐) ∩ 𝑘)) = if(((𝑒‘𝑐) ∩ 𝑑) = ∅, 𝑑, ((𝑒‘𝑐) ∩ 𝑑)))
9	4, 8	cbvmpov 7511	. . 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 2762	. . 3 ⊢ ∪ ran 𝑒 = ∪ ran 𝑒
11		seqomeq12 8446	. . 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 705	. 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 6882	. . . 4 ⊢ (𝑙 = 𝑦 → (𝑒‘𝑙) = (𝑒‘𝑦))
15	14	sseq2d 3966	. . 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 3424	. 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 2762	. 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 2762	. 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 2762	. 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 10349	1 ⊢ (𝐺 ∈ 𝐹 → ∃𝑓∀𝑏((𝑏:ω–1-1→V ∧ ∪ ran 𝑏 ⊆ 𝐺) → ((𝑓‘𝑏):ω–1-1→V ∧ ∪ ran (𝑓‘𝑏) ⊊ ∪ ran 𝑏)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2740  ∀wral 3078  {crab 3414  Vcvv 3453   ∖ cdif 3899   ∩ cin 3901   ⊆ wss 3902   ⊊ wpss 3903  ∅c0 4282  ifcif 4485  𝒫 cpw 4560  ∪ cuni 4870  ∩ cint 4910   class class class wbr 5107   ↦ cmpt 5190  ran crn 5660   ∘ ccom 5663  suc csuc 6363  –1-1→wf1 6534  ‘cfv 6537  ℩crio 7372  (class class class)co 7416   ∈ cmpo 7418  ωcom 7865  seq_ωcseqom 8439   ↑_m cmap 8829   ≈ cen 8952  Fincfn 8955

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-seqom 8440  df-1o 8458  df-er 8699  df-map 8831  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-card 9947

This theorem is used by:  fin23lem41  10357