Theorem fin23lem12 9742

Description: The beginning of the proof that every II-finite set (every chain of subsets has a maximal element) is III-finite (has no denumerable collection of subsets).

This first section is dedicated to the construction of 𝑈 and its intersection. First, the value of 𝑈 at a successor. (Contributed by Stefan O'Rear, 1-Nov-2014.)

Hypothesis

Ref	Expression
fin23lem.a	⊢ 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡)

Assertion

Ref	Expression
fin23lem12	⊢ (𝐴 ∈ ω → (𝑈‘suc 𝐴) = if(((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅, (𝑈‘𝐴), ((𝑡‘𝐴) ∩ (𝑈‘𝐴))))

Distinct variable groups:   𝑡,𝑖,𝑢   𝐴,𝑖,𝑢   𝑈,𝑖,𝑢

Allowed substitution hints:   𝐴(𝑡)   𝑈(𝑡)

Proof of Theorem fin23lem12

Step	Hyp	Ref	Expression
1		fin23lem.a	. . 3 ⊢ 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡)
2	1	seqomsuc 8076	. 2 ⊢ (𝐴 ∈ ω → (𝑈‘suc 𝐴) = (𝐴(𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢)))(𝑈‘𝐴)))
3		fvex 6658	. . 3 ⊢ (𝑈‘𝐴) ∈ V
4		fveq2 6645	. . . . . . 7 ⊢ (𝑖 = 𝐴 → (𝑡‘𝑖) = (𝑡‘𝐴))
5	4	ineq1d 4138	. . . . . 6 ⊢ (𝑖 = 𝐴 → ((𝑡‘𝑖) ∩ 𝑢) = ((𝑡‘𝐴) ∩ 𝑢))
6	5	eqeq1d 2800	. . . . 5 ⊢ (𝑖 = 𝐴 → (((𝑡‘𝑖) ∩ 𝑢) = ∅ ↔ ((𝑡‘𝐴) ∩ 𝑢) = ∅))
7	6, 5	ifbieq2d 4450	. . . 4 ⊢ (𝑖 = 𝐴 → if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢)) = if(((𝑡‘𝐴) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝐴) ∩ 𝑢)))
8		ineq2 4133	. . . . . 6 ⊢ (𝑢 = (𝑈‘𝐴) → ((𝑡‘𝐴) ∩ 𝑢) = ((𝑡‘𝐴) ∩ (𝑈‘𝐴)))
9	8	eqeq1d 2800	. . . . 5 ⊢ (𝑢 = (𝑈‘𝐴) → (((𝑡‘𝐴) ∩ 𝑢) = ∅ ↔ ((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅))
10		id 22	. . . . 5 ⊢ (𝑢 = (𝑈‘𝐴) → 𝑢 = (𝑈‘𝐴))
11	9, 10, 8	ifbieq12d 4452	. . . 4 ⊢ (𝑢 = (𝑈‘𝐴) → if(((𝑡‘𝐴) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝐴) ∩ 𝑢)) = if(((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅, (𝑈‘𝐴), ((𝑡‘𝐴) ∩ (𝑈‘𝐴))))
12		eqid 2798	. . . 4 ⊢ (𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))) = (𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢)))
13	3	inex2 5186	. . . . 5 ⊢ ((𝑡‘𝐴) ∩ (𝑈‘𝐴)) ∈ V
14	3, 13	ifex 4473	. . . 4 ⊢ if(((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅, (𝑈‘𝐴), ((𝑡‘𝐴) ∩ (𝑈‘𝐴))) ∈ V
15	7, 11, 12, 14	ovmpo 7289	. . 3 ⊢ ((𝐴 ∈ ω ∧ (𝑈‘𝐴) ∈ V) → (𝐴(𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢)))(𝑈‘𝐴)) = if(((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅, (𝑈‘𝐴), ((𝑡‘𝐴) ∩ (𝑈‘𝐴))))
16	3, 15	mpan2 690	. 2 ⊢ (𝐴 ∈ ω → (𝐴(𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢)))(𝑈‘𝐴)) = if(((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅, (𝑈‘𝐴), ((𝑡‘𝐴) ∩ (𝑈‘𝐴))))
17	2, 16	eqtrd 2833	1 ⊢ (𝐴 ∈ ω → (𝑈‘suc 𝐴) = if(((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅, (𝑈‘𝐴), ((𝑡‘𝐴) ∩ (𝑈‘𝐴))))

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ∩ cin 3880  ∅c0 4243  ifcif 4425  ∪ cuni 4800  ran crn 5520  suc csuc 6161  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137  ωcom 7560  seq_ωcseqom 8066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-seqom 8067

This theorem is referenced by:  fin23lem13  9743  fin23lem14  9744  fin23lem19  9747