Theorem omsmolem 8658

Description: Lemma for omsmo 8659. (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 2822	. . 3 ⊢ (𝑦 = ∅ → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ∅))
2		fveq2 6891	. . . 4 ⊢ (𝑦 = ∅ → (𝐹‘𝑦) = (𝐹‘∅))
3	2	eleq2d 2819	. . 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 2822	. . 3 ⊢ (𝑦 = 𝑤 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑤))
6		fveq2 6891	. . . 4 ⊢ (𝑦 = 𝑤 → (𝐹‘𝑦) = (𝐹‘𝑤))
7	6	eleq2d 2819	. . 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 2822	. . 3 ⊢ (𝑦 = suc 𝑤 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ suc 𝑤))
10		fveq2 6891	. . . 4 ⊢ (𝑦 = suc 𝑤 → (𝐹‘𝑦) = (𝐹‘suc 𝑤))
11	10	eleq2d 2819	. . 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 4330	. . . 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 3478	. . . . . 6 ⊢ 𝑧 ∈ V
17	16	elsuc 6434	. . . . 5 ⊢ (𝑧 ∈ suc 𝑤 ↔ (𝑧 ∈ 𝑤 ∨ 𝑧 = 𝑤))
18		fveq2 6891	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
19		suceq 6430	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → suc 𝑥 = suc 𝑤)
20	19	fveq2d 6895	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (𝐹‘suc 𝑥) = (𝐹‘suc 𝑤))
21	18, 20	eleq12d 2827	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥) ∈ (𝐹‘suc 𝑥) ↔ (𝐹‘𝑤) ∈ (𝐹‘suc 𝑤)))
22	21	rspccva 3611	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥) ∧ 𝑤 ∈ ω) → (𝐹‘𝑤) ∈ (𝐹‘suc 𝑤))
23	22	adantll 712	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) ∧ 𝑤 ∈ ω) → (𝐹‘𝑤) ∈ (𝐹‘suc 𝑤))
24		peano2b 7874	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ω ↔ suc 𝑤 ∈ ω)
25		ffvelcdm 7083	. . . . . . . . . . . . 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 3975	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ On → ((𝐹‘suc 𝑤) ∈ 𝐴 → (𝐹‘suc 𝑤) ∈ On))
28		ontr1 6410	. . . . . . . . . . . . 13 ⊢ ((𝐹‘suc 𝑤) ∈ On → (((𝐹‘𝑧) ∈ (𝐹‘𝑤) ∧ (𝐹‘𝑤) ∈ (𝐹‘suc 𝑤)) → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑤)))
29	28	expcomd 417	. . . . . . . . . . . 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 456	. . . . . . . . . 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 713	. . . . . . . . 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 407	. . . . . 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 6891	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (𝐹‘𝑧) = (𝐹‘𝑤))
37	36	eleq1d 2818	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → ((𝐹‘𝑧) ∈ (𝐹‘suc 𝑤) ↔ (𝐹‘𝑤) ∈ (𝐹‘suc 𝑤)))
38	22, 37	syl5ibrcom 246	. . . . . . 7 ⊢ ((∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥) ∧ 𝑤 ∈ ω) → (𝑧 = 𝑤 → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑤)))
39	38	ad4ant23 751	. . . . . 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 857	. . . . 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 241	. . . 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 420	. . 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 7893	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 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106  ∀wral 3061   ⊆ wss 3948  ∅c0 4322  Oncon0 6364  suc csuc 6366  ⟶wf 6539  ‘cfv 6543  ωcom 7857

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-om 7858

This theorem is referenced by:  omsmo  8659