Theorem omsmolem 8695

Description: Lemma for omsmo 8696. (Contributed by NM, 30-Nov-2003.) (Revised by David Abernethy, 1-Jan-2014.)

Assertion

Ref	Expression
omsmolem	⊢ (𝑦 ∈ ω → (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑧 ∈ 𝑦 → (𝐹‘𝑧) ∈ (𝐹‘𝑦))))

Distinct variable groups:   𝑦,𝑧,𝐴   𝑥,𝑦,𝑧,𝐹

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem omsmolem

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2830	. . 3 ⊢ (𝑦 = ∅ → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ∅))
2		fveq2 6906	. . . 4 ⊢ (𝑦 = ∅ → (𝐹‘𝑦) = (𝐹‘∅))
3	2	eleq2d 2827	. . 3 ⊢ (𝑦 = ∅ → ((𝐹‘𝑧) ∈ (𝐹‘𝑦) ↔ (𝐹‘𝑧) ∈ (𝐹‘∅)))
4	1, 3	imbi12d 344	. 2 ⊢ (𝑦 = ∅ → ((𝑧 ∈ 𝑦 → (𝐹‘𝑧) ∈ (𝐹‘𝑦)) ↔ (𝑧 ∈ ∅ → (𝐹‘𝑧) ∈ (𝐹‘∅))))
5		eleq2 2830	. . 3 ⊢ (𝑦 = 𝑤 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑤))
6		fveq2 6906	. . . 4 ⊢ (𝑦 = 𝑤 → (𝐹‘𝑦) = (𝐹‘𝑤))
7	6	eleq2d 2827	. . 3 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑧) ∈ (𝐹‘𝑦) ↔ (𝐹‘𝑧) ∈ (𝐹‘𝑤)))
8	5, 7	imbi12d 344	. 2 ⊢ (𝑦 = 𝑤 → ((𝑧 ∈ 𝑦 → (𝐹‘𝑧) ∈ (𝐹‘𝑦)) ↔ (𝑧 ∈ 𝑤 → (𝐹‘𝑧) ∈ (𝐹‘𝑤))))
9		eleq2 2830	. . 3 ⊢ (𝑦 = suc 𝑤 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ suc 𝑤))
10		fveq2 6906	. . . 4 ⊢ (𝑦 = suc 𝑤 → (𝐹‘𝑦) = (𝐹‘suc 𝑤))
11	10	eleq2d 2827	. . 3 ⊢ (𝑦 = suc 𝑤 → ((𝐹‘𝑧) ∈ (𝐹‘𝑦) ↔ (𝐹‘𝑧) ∈ (𝐹‘suc 𝑤)))
12	9, 11	imbi12d 344	. 2 ⊢ (𝑦 = suc 𝑤 → ((𝑧 ∈ 𝑦 → (𝐹‘𝑧) ∈ (𝐹‘𝑦)) ↔ (𝑧 ∈ suc 𝑤 → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑤))))
13		noel 4338	. . . 4 ⊢ ¬ 𝑧 ∈ ∅
14	13	pm2.21i 119	. . 3 ⊢ (𝑧 ∈ ∅ → (𝐹‘𝑧) ∈ (𝐹‘∅))
15	14	a1i 11	. 2 ⊢ (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑧 ∈ ∅ → (𝐹‘𝑧) ∈ (𝐹‘∅)))
16		vex 3484	. . . . . 6 ⊢ 𝑧 ∈ V
17	16	elsuc 6454	. . . . 5 ⊢ (𝑧 ∈ suc 𝑤 ↔ (𝑧 ∈ 𝑤 ∨ 𝑧 = 𝑤))
18		fveq2 6906	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
19		suceq 6450	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → suc 𝑥 = suc 𝑤)
20	19	fveq2d 6910	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (𝐹‘suc 𝑥) = (𝐹‘suc 𝑤))
21	18, 20	eleq12d 2835	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥) ∈ (𝐹‘suc 𝑥) ↔ (𝐹‘𝑤) ∈ (𝐹‘suc 𝑤)))
22	21	rspccva 3621	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥) ∧ 𝑤 ∈ ω) → (𝐹‘𝑤) ∈ (𝐹‘suc 𝑤))
23	22	adantll 714	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) ∧ 𝑤 ∈ ω) → (𝐹‘𝑤) ∈ (𝐹‘suc 𝑤))
24		peano2b 7904	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ω ↔ suc 𝑤 ∈ ω)
25		ffvelcdm 7101	. . . . . . . . . . . . 13 ⊢ ((𝐹:ω⟶𝐴 ∧ suc 𝑤 ∈ ω) → (𝐹‘suc 𝑤) ∈ 𝐴)
26	24, 25	sylan2b 594	. . . . . . . . . . . 12 ⊢ ((𝐹:ω⟶𝐴 ∧ 𝑤 ∈ ω) → (𝐹‘suc 𝑤) ∈ 𝐴)
27		ssel 3977	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ On → ((𝐹‘suc 𝑤) ∈ 𝐴 → (𝐹‘suc 𝑤) ∈ On))
28		ontr1 6430	. . . . . . . . . . . . 13 ⊢ ((𝐹‘suc 𝑤) ∈ On → (((𝐹‘𝑧) ∈ (𝐹‘𝑤) ∧ (𝐹‘𝑤) ∈ (𝐹‘suc 𝑤)) → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑤)))
29	28	expcomd 416	. . . . . . . . . . . 12 ⊢ ((𝐹‘suc 𝑤) ∈ On → ((𝐹‘𝑤) ∈ (𝐹‘suc 𝑤) → ((𝐹‘𝑧) ∈ (𝐹‘𝑤) → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑤))))
30	26, 27, 29	syl56 36	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ On → ((𝐹:ω⟶𝐴 ∧ 𝑤 ∈ ω) → ((𝐹‘𝑤) ∈ (𝐹‘suc 𝑤) → ((𝐹‘𝑧) ∈ (𝐹‘𝑤) → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑤)))))
31	30	impl 455	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ 𝑤 ∈ ω) → ((𝐹‘𝑤) ∈ (𝐹‘suc 𝑤) → ((𝐹‘𝑧) ∈ (𝐹‘𝑤) → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑤))))
32	31	adantlr 715	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) ∧ 𝑤 ∈ ω) → ((𝐹‘𝑤) ∈ (𝐹‘suc 𝑤) → ((𝐹‘𝑧) ∈ (𝐹‘𝑤) → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑤))))
33	23, 32	mpd 15	. . . . . . . 8 ⊢ ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) ∧ 𝑤 ∈ ω) → ((𝐹‘𝑧) ∈ (𝐹‘𝑤) → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑤)))
34	33	imim2d 57	. . . . . . 7 ⊢ ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) ∧ 𝑤 ∈ ω) → ((𝑧 ∈ 𝑤 → (𝐹‘𝑧) ∈ (𝐹‘𝑤)) → (𝑧 ∈ 𝑤 → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑤))))
35	34	imp 406	. . . . . 6 ⊢ (((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) ∧ 𝑤 ∈ ω) ∧ (𝑧 ∈ 𝑤 → (𝐹‘𝑧) ∈ (𝐹‘𝑤))) → (𝑧 ∈ 𝑤 → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑤)))
36		fveq2 6906	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (𝐹‘𝑧) = (𝐹‘𝑤))
37	36	eleq1d 2826	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → ((𝐹‘𝑧) ∈ (𝐹‘suc 𝑤) ↔ (𝐹‘𝑤) ∈ (𝐹‘suc 𝑤)))
38	22, 37	syl5ibrcom 247	. . . . . . 7 ⊢ ((∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥) ∧ 𝑤 ∈ ω) → (𝑧 = 𝑤 → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑤)))
39	38	ad4ant23 753	. . . . . 6 ⊢ (((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) ∧ 𝑤 ∈ ω) ∧ (𝑧 ∈ 𝑤 → (𝐹‘𝑧) ∈ (𝐹‘𝑤))) → (𝑧 = 𝑤 → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑤)))
40	35, 39	jaod 860	. . . . 5 ⊢ (((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) ∧ 𝑤 ∈ ω) ∧ (𝑧 ∈ 𝑤 → (𝐹‘𝑧) ∈ (𝐹‘𝑤))) → ((𝑧 ∈ 𝑤 ∨ 𝑧 = 𝑤) → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑤)))
41	17, 40	biimtrid 242	. . . 4 ⊢ (((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) ∧ 𝑤 ∈ ω) ∧ (𝑧 ∈ 𝑤 → (𝐹‘𝑧) ∈ (𝐹‘𝑤))) → (𝑧 ∈ suc 𝑤 → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑤)))
42	41	exp31 419	. . 3 ⊢ (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑤 ∈ ω → ((𝑧 ∈ 𝑤 → (𝐹‘𝑧) ∈ (𝐹‘𝑤)) → (𝑧 ∈ suc 𝑤 → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑤)))))
43	42	com12 32	. 2 ⊢ (𝑤 ∈ ω → (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) → ((𝑧 ∈ 𝑤 → (𝐹‘𝑧) ∈ (𝐹‘𝑤)) → (𝑧 ∈ suc 𝑤 → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑤)))))
44	4, 8, 12, 15, 43	finds2 7920	1 ⊢ (𝑦 ∈ ω → (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑧 ∈ 𝑦 → (𝐹‘𝑧) ∈ (𝐹‘𝑦))))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   = wceq 1540   ∈ wcel 2108  ∀wral 3061   ⊆ wss 3951  ∅c0 4333  Oncon0 6384  suc csuc 6386  ⟶wf 6557  ‘cfv 6561  ωcom 7887

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-om 7888

This theorem is referenced by:  omsmo  8696