Theorem fin23lem12 10291

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 8428	. 2 ⊢ (𝐴 ∈ ω → (𝑈‘suc 𝐴) = (𝐴(𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢)))(𝑈‘𝐴)))
3		fvex 6874	. . 3 ⊢ (𝑈‘𝐴) ∈ V
4		fveq2 6861	. . . . . . 7 ⊢ (𝑖 = 𝐴 → (𝑡‘𝑖) = (𝑡‘𝐴))
5	4	ineq1d 4185	. . . . . 6 ⊢ (𝑖 = 𝐴 → ((𝑡‘𝑖) ∩ 𝑢) = ((𝑡‘𝐴) ∩ 𝑢))
6	5	eqeq1d 2732	. . . . 5 ⊢ (𝑖 = 𝐴 → (((𝑡‘𝑖) ∩ 𝑢) = ∅ ↔ ((𝑡‘𝐴) ∩ 𝑢) = ∅))
7	6, 5	ifbieq2d 4518	. . . 4 ⊢ (𝑖 = 𝐴 → if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢)) = if(((𝑡‘𝐴) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝐴) ∩ 𝑢)))
8		ineq2 4180	. . . . . 6 ⊢ (𝑢 = (𝑈‘𝐴) → ((𝑡‘𝐴) ∩ 𝑢) = ((𝑡‘𝐴) ∩ (𝑈‘𝐴)))
9	8	eqeq1d 2732	. . . . 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 4520	. . . 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 2730	. . . 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 5276	. . . . 5 ⊢ ((𝑡‘𝐴) ∩ (𝑈‘𝐴)) ∈ V
14	3, 13	ifex 4542	. . . 4 ⊢ if(((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅, (𝑈‘𝐴), ((𝑡‘𝐴) ∩ (𝑈‘𝐴))) ∈ V
15	7, 11, 12, 14	ovmpo 7552	. . 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 691	. 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 2765	1 ⊢ (𝐴 ∈ ω → (𝑈‘suc 𝐴) = if(((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅, (𝑈‘𝐴), ((𝑡‘𝐴) ∩ (𝑈‘𝐴))))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3450   ∩ cin 3916  ∅c0 4299  ifcif 4491  ∪ cuni 4874  ran crn 5642  suc csuc 6337  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392  ωcom 7845  seq_ωcseqom 8418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-seqom 8419

This theorem is referenced by:  fin23lem13  10292  fin23lem14  10293  fin23lem19  10296