Theorem unblem3 8839

Description: Lemma for unbnn 8841. 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 8838	. . . . . 6 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → (𝑧 ∈ ω → (𝐹‘𝑧) ∈ 𝐴))
3	2	imp 410	. . . . 5 ⊢ (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) ∧ 𝑧 ∈ ω) → (𝐹‘𝑧) ∈ 𝐴)
4		omsson 7597	. . . . . . . 8 ⊢ ω ⊆ On
5		sstr 3883	. . . . . . . 8 ⊢ ((𝐴 ⊆ ω ∧ ω ⊆ On) → 𝐴 ⊆ On)
6	4, 5	mpan2 691	. . . . . . 7 ⊢ (𝐴 ⊆ ω → 𝐴 ⊆ On)
7		ssel 3868	. . . . . . . 8 ⊢ (𝐴 ⊆ On → ((𝐹‘𝑧) ∈ 𝐴 → (𝐹‘𝑧) ∈ On))
8	7	anc2li 559	. . . . . . 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 726	. . . . 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 6255	. . . 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 8837	. . . . . . 7 ⊢ (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) ∧ (𝐹‘𝑧) ∈ 𝐴) → ∩ (𝐴 ∖ suc (𝐹‘𝑧)) ∈ 𝐴)
15	14	ex 416	. . . . . 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 6231	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → suc 𝑦 = suc 𝑥)
18	17	difeq2d 4011	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc 𝑥))
19	18	inteqd 4838	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ∩ (𝐴 ∖ suc 𝑦) = ∩ (𝐴 ∖ suc 𝑥))
20		suceq 6231	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑧) → suc 𝑦 = suc (𝐹‘𝑧))
21	20	difeq2d 4011	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑧) → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc (𝐹‘𝑧)))
22	21	inteqd 4838	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑧) → ∩ (𝐴 ∖ suc 𝑦) = ∩ (𝐴 ∖ suc (𝐹‘𝑧)))
23	1, 19, 22	frsucmpt2 8098	. . . . . 6 ⊢ ((𝑧 ∈ ω ∧ ∩ (𝐴 ∖ suc (𝐹‘𝑧)) ∈ 𝐴) → (𝐹‘suc 𝑧) = ∩ (𝐴 ∖ suc (𝐹‘𝑧)))
24	23	ex 416	. . . . 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 410	. . 3 ⊢ (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) ∧ 𝑧 ∈ ω) → (𝐹‘suc 𝑧) = ∩ (𝐴 ∖ suc (𝐹‘𝑧)))
27	13, 26	eleqtrrd 2836	. 2 ⊢ (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) ∧ 𝑧 ∈ ω) → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑧))
28	27	ex 416	1 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → (𝑧 ∈ ω → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑧)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1542   ∈ wcel 2113  ∀wral 3053  ∃wrex 3054  Vcvv 3397   ∖ cdif 3838   ⊆ wss 3841  ∩ cint 4833   ↦ cmpt 5107   ↾ cres 5521  Oncon0 6166  suc csuc 6168  ‘cfv 6333  ωcom 7593  reccrdg 8067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293  ax-un 7473

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-pss 3860  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-tp 4518  df-op 4520  df-uni 4794  df-int 4834  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-tr 5134  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 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-om 7594  df-wrecs 7969  df-recs 8030  df-rdg 8068

This theorem is referenced by:  unblem4  8840