Theorem fin23lem12 10247

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 8390	. 2 ⊢ (𝐴 ∈ ω → (𝑈‘suc 𝐴) = (𝐴(𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢)))(𝑈‘𝐴)))
3		fvex 6848	. . 3 ⊢ (𝑈‘𝐴) ∈ V
4		fveq2 6835	. . . . . . 7 ⊢ (𝑖 = 𝐴 → (𝑡‘𝑖) = (𝑡‘𝐴))
5	4	ineq1d 4160	. . . . . 6 ⊢ (𝑖 = 𝐴 → ((𝑡‘𝑖) ∩ 𝑢) = ((𝑡‘𝐴) ∩ 𝑢))
6	5	eqeq1d 2739	. . . . 5 ⊢ (𝑖 = 𝐴 → (((𝑡‘𝑖) ∩ 𝑢) = ∅ ↔ ((𝑡‘𝐴) ∩ 𝑢) = ∅))
7	6, 5	ifbieq2d 4494	. . . 4 ⊢ (𝑖 = 𝐴 → if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢)) = if(((𝑡‘𝐴) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝐴) ∩ 𝑢)))
8		ineq2 4155	. . . . . 6 ⊢ (𝑢 = (𝑈‘𝐴) → ((𝑡‘𝐴) ∩ 𝑢) = ((𝑡‘𝐴) ∩ (𝑈‘𝐴)))
9	8	eqeq1d 2739	. . . . 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 4496	. . . 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 2737	. . . 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 5256	. . . . 5 ⊢ ((𝑡‘𝐴) ∩ (𝑈‘𝐴)) ∈ V
14	3, 13	ifex 4518	. . . 4 ⊢ if(((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅, (𝑈‘𝐴), ((𝑡‘𝐴) ∩ (𝑈‘𝐴))) ∈ V
15	7, 11, 12, 14	ovmpo 7521	. . 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 692	. 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 2772	1 ⊢ (𝐴 ∈ ω → (𝑈‘suc 𝐴) = if(((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅, (𝑈‘𝐴), ((𝑡‘𝐴) ∩ (𝑈‘𝐴))))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3430   ∩ cin 3889  ∅c0 4274  ifcif 4467  ∪ cuni 4851  ran crn 5626  suc csuc 6320  ‘cfv 6493  (class class class)co 7361   ∈ cmpo 7363  ωcom 7811  seq_ωcseqom 8380

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-seqom 8381

This theorem is referenced by:  fin23lem13  10248  fin23lem14  10249  fin23lem19  10252