Theorem unblem3 9255

Description: Lemma for unbnn 9257. The value of the function 𝐹 is less than its value at a successor. (Contributed by NM, 3-Dec-2003.)

Hypothesis

Ref	Expression
unblem.2	⊢ 𝐹 = (rec((𝑥 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑥)), ∩ 𝐴) ↾ ω)

Assertion

Ref	Expression
unblem3	⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → (𝑧 ∈ ω → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑧)))

Distinct variable groups:   𝑤,𝑣,𝑥,𝑧,𝐴   𝑣,𝐹,𝑤,𝑧

Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem unblem3

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		unblem.2	. . . . . . 7 ⊢ 𝐹 = (rec((𝑥 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑥)), ∩ 𝐴) ↾ ω)
2	1	unblem2 9254	. . . . . 6 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → (𝑧 ∈ ω → (𝐹‘𝑧) ∈ 𝐴))
3	2	imp 411	. . . . 5 ⊢ (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) ∧ 𝑧 ∈ ω) → (𝐹‘𝑧) ∈ 𝐴)
4		omsson 7867	. . . . . . . 8 ⊢ ω ⊆ On
5		sstr 3946	. . . . . . . 8 ⊢ ((𝐴 ⊆ ω ∧ ω ⊆ On) → 𝐴 ⊆ On)
6	4, 5	mpan2 703	. . . . . . 7 ⊢ (𝐴 ⊆ ω → 𝐴 ⊆ On)
7		ssel 3932	. . . . . . . 8 ⊢ (𝐴 ⊆ On → ((𝐹‘𝑧) ∈ 𝐴 → (𝐹‘𝑧) ∈ On))
8	7	anc2li 564	. . . . . . 7 ⊢ (𝐴 ⊆ On → ((𝐹‘𝑧) ∈ 𝐴 → (𝐴 ⊆ On ∧ (𝐹‘𝑧) ∈ On)))
9	6, 8	syl 18	. . . . . 6 ⊢ (𝐴 ⊆ ω → ((𝐹‘𝑧) ∈ 𝐴 → (𝐴 ⊆ On ∧ (𝐹‘𝑧) ∈ On)))
10	9	ad2antrr 738	. . . . 5 ⊢ (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) ∧ 𝑧 ∈ ω) → ((𝐹‘𝑧) ∈ 𝐴 → (𝐴 ⊆ On ∧ (𝐹‘𝑧) ∈ On)))
11	3, 10	mpd 16	. . . 4 ⊢ (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) ∧ 𝑧 ∈ ω) → (𝐴 ⊆ On ∧ (𝐹‘𝑧) ∈ On))
12		onmindif 6457	. . . 4 ⊢ ((𝐴 ⊆ On ∧ (𝐹‘𝑧) ∈ On) → (𝐹‘𝑧) ∈ ∩ (𝐴 ∖ suc (𝐹‘𝑧)))
13	11, 12	syl 18	. . 3 ⊢ (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) ∧ 𝑧 ∈ ω) → (𝐹‘𝑧) ∈ ∩ (𝐴 ∖ suc (𝐹‘𝑧)))
14		unblem1 9253	. . . . . . 7 ⊢ (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) ∧ (𝐹‘𝑧) ∈ 𝐴) → ∩ (𝐴 ∖ suc (𝐹‘𝑧)) ∈ 𝐴)
15	14	ex 417	. . . . . 6 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → ((𝐹‘𝑧) ∈ 𝐴 → ∩ (𝐴 ∖ suc (𝐹‘𝑧)) ∈ 𝐴))
16	2, 15	syld 48	. . . . 5 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → (𝑧 ∈ ω → ∩ (𝐴 ∖ suc (𝐹‘𝑧)) ∈ 𝐴))
17		suceq 6431	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → suc 𝑦 = suc 𝑥)
18	17	difeq2d 4082	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc 𝑥))
19	18	inteqd 4918	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ∩ (𝐴 ∖ suc 𝑦) = ∩ (𝐴 ∖ suc 𝑥))
20		suceq 6431	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑧) → suc 𝑦 = suc (𝐹‘𝑧))
21	20	difeq2d 4082	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑧) → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc (𝐹‘𝑧)))
22	21	inteqd 4918	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑧) → ∩ (𝐴 ∖ suc 𝑦) = ∩ (𝐴 ∖ suc (𝐹‘𝑧)))
23	1, 19, 22	frsucmpt2 8428	. . . . . 6 ⊢ ((𝑧 ∈ ω ∧ ∩ (𝐴 ∖ suc (𝐹‘𝑧)) ∈ 𝐴) → (𝐹‘suc 𝑧) = ∩ (𝐴 ∖ suc (𝐹‘𝑧)))
24	23	ex 417	. . . . 5 ⊢ (𝑧 ∈ ω → (∩ (𝐴 ∖ suc (𝐹‘𝑧)) ∈ 𝐴 → (𝐹‘suc 𝑧) = ∩ (𝐴 ∖ suc (𝐹‘𝑧))))
25	16, 24	sylcom 31	. . . 4 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → (𝑧 ∈ ω → (𝐹‘suc 𝑧) = ∩ (𝐴 ∖ suc (𝐹‘𝑧))))
26	25	imp 411	. . 3 ⊢ (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) ∧ 𝑧 ∈ ω) → (𝐹‘suc 𝑧) = ∩ (𝐴 ∖ suc (𝐹‘𝑧)))
27	13, 26	eleqtrrd 2866	. 2 ⊢ (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) ∧ 𝑧 ∈ ω) → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑧))
28	27	ex 417	1 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → (𝑧 ∈ ω → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑧)))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089  Vcvv 3455   ∖ cdif 3903   ⊆ wss 3906  ∩ cint 4913   ↦ cmpt 5193   ↾ cres 5665  Oncon0 6362  suc csuc 6364  ‘cfv 6538  ωcom 7863  reccrdg 8397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-om 7864  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398

This theorem is referenced by:  unblem4  9256