Theorem unblem3 9328

Description: Lemma for unbnn 9330. 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 9327	. . . . . 6 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → (𝑧 ∈ ω → (𝐹‘𝑧) ∈ 𝐴))
3	2	imp 405	. . . . 5 ⊢ (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) ∧ 𝑧 ∈ ω) → (𝐹‘𝑧) ∈ 𝐴)
4		omsson 7880	. . . . . . . 8 ⊢ ω ⊆ On
5		sstr 3990	. . . . . . . 8 ⊢ ((𝐴 ⊆ ω ∧ ω ⊆ On) → 𝐴 ⊆ On)
6	4, 5	mpan2 689	. . . . . . 7 ⊢ (𝐴 ⊆ ω → 𝐴 ⊆ On)
7		ssel 3975	. . . . . . . 8 ⊢ (𝐴 ⊆ On → ((𝐹‘𝑧) ∈ 𝐴 → (𝐹‘𝑧) ∈ On))
8	7	anc2li 554	. . . . . . 7 ⊢ (𝐴 ⊆ On → ((𝐹‘𝑧) ∈ 𝐴 → (𝐴 ⊆ On ∧ (𝐹‘𝑧) ∈ On)))
9	6, 8	syl 17	. . . . . 6 ⊢ (𝐴 ⊆ ω → ((𝐹‘𝑧) ∈ 𝐴 → (𝐴 ⊆ On ∧ (𝐹‘𝑧) ∈ On)))
10	9	ad2antrr 724	. . . . 5 ⊢ (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) ∧ 𝑧 ∈ ω) → ((𝐹‘𝑧) ∈ 𝐴 → (𝐴 ⊆ On ∧ (𝐹‘𝑧) ∈ On)))
11	3, 10	mpd 15	. . . 4 ⊢ (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) ∧ 𝑧 ∈ ω) → (𝐴 ⊆ On ∧ (𝐹‘𝑧) ∈ On))
12		onmindif 6466	. . . 4 ⊢ ((𝐴 ⊆ On ∧ (𝐹‘𝑧) ∈ On) → (𝐹‘𝑧) ∈ ∩ (𝐴 ∖ suc (𝐹‘𝑧)))
13	11, 12	syl 17	. . 3 ⊢ (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) ∧ 𝑧 ∈ ω) → (𝐹‘𝑧) ∈ ∩ (𝐴 ∖ suc (𝐹‘𝑧)))
14		unblem1 9326	. . . . . . 7 ⊢ (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) ∧ (𝐹‘𝑧) ∈ 𝐴) → ∩ (𝐴 ∖ suc (𝐹‘𝑧)) ∈ 𝐴)
15	14	ex 411	. . . . . 6 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → ((𝐹‘𝑧) ∈ 𝐴 → ∩ (𝐴 ∖ suc (𝐹‘𝑧)) ∈ 𝐴))
16	2, 15	syld 47	. . . . 5 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → (𝑧 ∈ ω → ∩ (𝐴 ∖ suc (𝐹‘𝑧)) ∈ 𝐴))
17		suceq 6440	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → suc 𝑦 = suc 𝑥)
18	17	difeq2d 4122	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc 𝑥))
19	18	inteqd 4958	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ∩ (𝐴 ∖ suc 𝑦) = ∩ (𝐴 ∖ suc 𝑥))
20		suceq 6440	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑧) → suc 𝑦 = suc (𝐹‘𝑧))
21	20	difeq2d 4122	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑧) → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc (𝐹‘𝑧)))
22	21	inteqd 4958	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑧) → ∩ (𝐴 ∖ suc 𝑦) = ∩ (𝐴 ∖ suc (𝐹‘𝑧)))
23	1, 19, 22	frsucmpt2 8467	. . . . . 6 ⊢ ((𝑧 ∈ ω ∧ ∩ (𝐴 ∖ suc (𝐹‘𝑧)) ∈ 𝐴) → (𝐹‘suc 𝑧) = ∩ (𝐴 ∖ suc (𝐹‘𝑧)))
24	23	ex 411	. . . . 5 ⊢ (𝑧 ∈ ω → (∩ (𝐴 ∖ suc (𝐹‘𝑧)) ∈ 𝐴 → (𝐹‘suc 𝑧) = ∩ (𝐴 ∖ suc (𝐹‘𝑧))))
25	16, 24	sylcom 30	. . . 4 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → (𝑧 ∈ ω → (𝐹‘suc 𝑧) = ∩ (𝐴 ∖ suc (𝐹‘𝑧))))
26	25	imp 405	. . 3 ⊢ (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) ∧ 𝑧 ∈ ω) → (𝐹‘suc 𝑧) = ∩ (𝐴 ∖ suc (𝐹‘𝑧)))
27	13, 26	eleqtrrd 2832	. 2 ⊢ (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) ∧ 𝑧 ∈ ω) → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑧))
28	27	ex 411	1 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → (𝑧 ∈ ω → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑧)))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∀wral 3058  ∃wrex 3067  Vcvv 3473   ∖ cdif 3946   ⊆ wss 3949  ∩ cint 4953   ↦ cmpt 5235   ↾ cres 5684  Oncon0 6374  suc csuc 6376  ‘cfv 6553  ωcom 7876  reccrdg 8436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-om 7877  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437

This theorem is referenced by:  unblem4  9329