Theorem omsmolem 8644

Description: Lemma for omsmo 8645. (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 2849	. . 3 ⊢ (𝑦 = ∅ → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ∅))
2		fveq2 6873	. . . 4 ⊢ (𝑦 = ∅ → (𝐹‘𝑦) = (𝐹‘∅))
3	2	eleq2d 2846	. . 3 ⊢ (𝑦 = ∅ → ((𝐹‘𝑧) ∈ (𝐹‘𝑦) ↔ (𝐹‘𝑧) ∈ (𝐹‘∅)))
4	1, 3	imbi12d 347	. 2 ⊢ (𝑦 = ∅ → ((𝑧 ∈ 𝑦 → (𝐹‘𝑧) ∈ (𝐹‘𝑦)) ↔ (𝑧 ∈ ∅ → (𝐹‘𝑧) ∈ (𝐹‘∅))))
5		eleq2 2849	. . 3 ⊢ (𝑦 = 𝑤 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑤))
6		fveq2 6873	. . . 4 ⊢ (𝑦 = 𝑤 → (𝐹‘𝑦) = (𝐹‘𝑤))
7	6	eleq2d 2846	. . 3 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑧) ∈ (𝐹‘𝑦) ↔ (𝐹‘𝑧) ∈ (𝐹‘𝑤)))
8	5, 7	imbi12d 347	. 2 ⊢ (𝑦 = 𝑤 → ((𝑧 ∈ 𝑦 → (𝐹‘𝑧) ∈ (𝐹‘𝑦)) ↔ (𝑧 ∈ 𝑤 → (𝐹‘𝑧) ∈ (𝐹‘𝑤))))
9		eleq2 2849	. . 3 ⊢ (𝑦 = suc 𝑤 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ suc 𝑤))
10		fveq2 6873	. . . 4 ⊢ (𝑦 = suc 𝑤 → (𝐹‘𝑦) = (𝐹‘suc 𝑤))
11	10	eleq2d 2846	. . 3 ⊢ (𝑦 = suc 𝑤 → ((𝐹‘𝑧) ∈ (𝐹‘𝑦) ↔ (𝐹‘𝑧) ∈ (𝐹‘suc 𝑤)))
12	9, 11	imbi12d 347	. 2 ⊢ (𝑦 = suc 𝑤 → ((𝑧 ∈ 𝑦 → (𝐹‘𝑧) ∈ (𝐹‘𝑦)) ↔ (𝑧 ∈ suc 𝑤 → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑤))))
13		noel 4283	. . . 4 ⊢ ¬ 𝑧 ∈ ∅
14	13	pm2.21i 120	. . 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 3454	. . . . . 6 ⊢ 𝑧 ∈ V
17	16	elsuc 6424	. . . . 5 ⊢ (𝑧 ∈ suc 𝑤 ↔ (𝑧 ∈ 𝑤 ∨ 𝑧 = 𝑤))
18		fveq2 6873	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
19		suceq 6420	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → suc 𝑥 = suc 𝑤)
20	19	fveq2d 6877	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (𝐹‘suc 𝑥) = (𝐹‘suc 𝑤))
21	18, 20	eleq12d 2854	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥) ∈ (𝐹‘suc 𝑥) ↔ (𝐹‘𝑤) ∈ (𝐹‘suc 𝑤)))
22	21	rspccva 3575	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥) ∧ 𝑤 ∈ ω) → (𝐹‘𝑤) ∈ (𝐹‘suc 𝑤))
23	22	adantll 727	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) ∧ 𝑤 ∈ ω) → (𝐹‘𝑤) ∈ (𝐹‘suc 𝑤))
24		peano2b 7877	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ω ↔ suc 𝑤 ∈ ω)
25		ffvelcdm 7069	. . . . . . . . . . . . 13 ⊢ ((𝐹:ω⟶𝐴 ∧ suc 𝑤 ∈ ω) → (𝐹‘suc 𝑤) ∈ 𝐴)
26	24, 25	sylan2b 606	. . . . . . . . . . . 12 ⊢ ((𝐹:ω⟶𝐴 ∧ 𝑤 ∈ ω) → (𝐹‘suc 𝑤) ∈ 𝐴)
27		ssel 3924	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ On → ((𝐹‘suc 𝑤) ∈ 𝐴 → (𝐹‘suc 𝑤) ∈ On))
28		ontr1 6399	. . . . . . . . . . . . 13 ⊢ ((𝐹‘suc 𝑤) ∈ On → (((𝐹‘𝑧) ∈ (𝐹‘𝑤) ∧ (𝐹‘𝑤) ∈ (𝐹‘suc 𝑤)) → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑤)))
29	28	expcomd 422	. . . . . . . . . . . 12 ⊢ ((𝐹‘suc 𝑤) ∈ On → ((𝐹‘𝑤) ∈ (𝐹‘suc 𝑤) → ((𝐹‘𝑧) ∈ (𝐹‘𝑤) → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑤))))
30	26, 27, 29	syl56 37	. . . . . . . . . . 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 461	. . . . . . . . . 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 728	. . . . . . . . 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 16	. . . . . . . 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 58	. . . . . . 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 412	. . . . . 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 6873	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (𝐹‘𝑧) = (𝐹‘𝑤))
37	36	eleq1d 2845	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → ((𝐹‘𝑧) ∈ (𝐹‘suc 𝑤) ↔ (𝐹‘𝑤) ∈ (𝐹‘suc 𝑤)))
38	22, 37	syl5ibrcom 250	. . . . . . 7 ⊢ ((∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥) ∧ 𝑤 ∈ ω) → (𝑧 = 𝑤 → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑤)))
39	38	ad4ant23 766	. . . . . 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 873	. . . . 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 245	. . . 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 425	. . 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 33	. 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))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∀wral 3076   ⊆ wss 3898  ∅c0 4278  Oncon0 6351  suc csuc 6353  ⟶wf 6523  ‘cfv 6527  ωcom 7860

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-fv 6535  df-om 7861

This theorem is used by:  omsmo  8645