Theorem unblem3 9197

Description: Lemma for unbnn 9199. 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 9196	. . . . . 6 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → (𝑧 ∈ ω → (𝐹‘𝑧) ∈ 𝐴))
3	2	imp 406	. . . . 5 ⊢ (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) ∧ 𝑧 ∈ ω) → (𝐹‘𝑧) ∈ 𝐴)
4		omsson 7814	. . . . . . . 8 ⊢ ω ⊆ On
5		sstr 3931	. . . . . . . 8 ⊢ ((𝐴 ⊆ ω ∧ ω ⊆ On) → 𝐴 ⊆ On)
6	4, 5	mpan2 692	. . . . . . 7 ⊢ (𝐴 ⊆ ω → 𝐴 ⊆ On)
7		ssel 3916	. . . . . . . 8 ⊢ (𝐴 ⊆ On → ((𝐹‘𝑧) ∈ 𝐴 → (𝐹‘𝑧) ∈ On))
8	7	anc2li 555	. . . . . . 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 727	. . . . 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 6411	. . . 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 9195	. . . . . . 7 ⊢ (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) ∧ (𝐹‘𝑧) ∈ 𝐴) → ∩ (𝐴 ∖ suc (𝐹‘𝑧)) ∈ 𝐴)
15	14	ex 412	. . . . . 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 6385	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → suc 𝑦 = suc 𝑥)
18	17	difeq2d 4067	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc 𝑥))
19	18	inteqd 4895	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ∩ (𝐴 ∖ suc 𝑦) = ∩ (𝐴 ∖ suc 𝑥))
20		suceq 6385	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑧) → suc 𝑦 = suc (𝐹‘𝑧))
21	20	difeq2d 4067	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑧) → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc (𝐹‘𝑧)))
22	21	inteqd 4895	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑧) → ∩ (𝐴 ∖ suc 𝑦) = ∩ (𝐴 ∖ suc (𝐹‘𝑧)))
23	1, 19, 22	frsucmpt2 8372	. . . . . 6 ⊢ ((𝑧 ∈ ω ∧ ∩ (𝐴 ∖ suc (𝐹‘𝑧)) ∈ 𝐴) → (𝐹‘suc 𝑧) = ∩ (𝐴 ∖ suc (𝐹‘𝑧)))
24	23	ex 412	. . . . 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 406	. . 3 ⊢ (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) ∧ 𝑧 ∈ ω) → (𝐹‘suc 𝑧) = ∩ (𝐴 ∖ suc (𝐹‘𝑧)))
27	13, 26	eleqtrrd 2840	. 2 ⊢ (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) ∧ 𝑧 ∈ ω) → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑧))
28	27	ex 412	1 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → (𝑧 ∈ ω → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑧)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  Vcvv 3430   ∖ cdif 3887   ⊆ wss 3890  ∩ cint 4890   ↦ cmpt 5167   ↾ cres 5626  Oncon0 6317  suc csuc 6319  ‘cfv 6492  ωcom 7810  reccrdg 8341

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 5231  ax-nul 5241  ax-pr 5370  ax-un 7682

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-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-om 7811  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342

This theorem is referenced by:  unblem4  9198