Theorem omsmolem 8627

Description: Lemma for omsmo 8628. (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 2852	. . 3 ⊢ (𝑦 = ∅ → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ∅))
2		fveq2 6867	. . . 4 ⊢ (𝑦 = ∅ → (𝐹‘𝑦) = (𝐹‘∅))
3	2	eleq2d 2849	. . 3 ⊢ (𝑦 = ∅ → ((𝐹‘𝑧) ∈ (𝐹‘𝑦) ↔ (𝐹‘𝑧) ∈ (𝐹‘∅)))
4	1, 3	imbi12d 346	. 2 ⊢ (𝑦 = ∅ → ((𝑧 ∈ 𝑦 → (𝐹‘𝑧) ∈ (𝐹‘𝑦)) ↔ (𝑧 ∈ ∅ → (𝐹‘𝑧) ∈ (𝐹‘∅))))
5		eleq2 2852	. . 3 ⊢ (𝑦 = 𝑤 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑤))
6		fveq2 6867	. . . 4 ⊢ (𝑦 = 𝑤 → (𝐹‘𝑦) = (𝐹‘𝑤))
7	6	eleq2d 2849	. . 3 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑧) ∈ (𝐹‘𝑦) ↔ (𝐹‘𝑧) ∈ (𝐹‘𝑤)))
8	5, 7	imbi12d 346	. 2 ⊢ (𝑦 = 𝑤 → ((𝑧 ∈ 𝑦 → (𝐹‘𝑧) ∈ (𝐹‘𝑦)) ↔ (𝑧 ∈ 𝑤 → (𝐹‘𝑧) ∈ (𝐹‘𝑤))))
9		eleq2 2852	. . 3 ⊢ (𝑦 = suc 𝑤 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ suc 𝑤))
10		fveq2 6867	. . . 4 ⊢ (𝑦 = suc 𝑤 → (𝐹‘𝑦) = (𝐹‘suc 𝑤))
11	10	eleq2d 2849	. . 3 ⊢ (𝑦 = suc 𝑤 → ((𝐹‘𝑧) ∈ (𝐹‘𝑦) ↔ (𝐹‘𝑧) ∈ (𝐹‘suc 𝑤)))
12	9, 11	imbi12d 346	. 2 ⊢ (𝑦 = suc 𝑤 → ((𝑧 ∈ 𝑦 → (𝐹‘𝑧) ∈ (𝐹‘𝑦)) ↔ (𝑧 ∈ suc 𝑤 → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑤))))
13		noel 4291	. . . 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 3459	. . . . . 6 ⊢ 𝑧 ∈ V
17	16	elsuc 6418	. . . . 5 ⊢ (𝑧 ∈ suc 𝑤 ↔ (𝑧 ∈ 𝑤 ∨ 𝑧 = 𝑤))
18		fveq2 6867	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
19		suceq 6414	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → suc 𝑥 = suc 𝑤)
20	19	fveq2d 6871	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (𝐹‘suc 𝑥) = (𝐹‘suc 𝑤))
21	18, 20	eleq12d 2857	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥) ∈ (𝐹‘suc 𝑥) ↔ (𝐹‘𝑤) ∈ (𝐹‘suc 𝑤)))
22	21	rspccva 3581	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥) ∧ 𝑤 ∈ ω) → (𝐹‘𝑤) ∈ (𝐹‘suc 𝑤))
23	22	adantll 724	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) ∧ 𝑤 ∈ ω) → (𝐹‘𝑤) ∈ (𝐹‘suc 𝑤))
24		peano2b 7863	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ω ↔ suc 𝑤 ∈ ω)
25		ffvelcdm 7062	. . . . . . . . . . . . 13 ⊢ ((𝐹:ω⟶𝐴 ∧ suc 𝑤 ∈ ω) → (𝐹‘suc 𝑤) ∈ 𝐴)
26	24, 25	sylan2b 603	. . . . . . . . . . . 12 ⊢ ((𝐹:ω⟶𝐴 ∧ 𝑤 ∈ ω) → (𝐹‘suc 𝑤) ∈ 𝐴)
27		ssel 3931	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ On → ((𝐹‘suc 𝑤) ∈ 𝐴 → (𝐹‘suc 𝑤) ∈ On))
28		ontr1 6393	. . . . . . . . . . . . 13 ⊢ ((𝐹‘suc 𝑤) ∈ On → (((𝐹‘𝑧) ∈ (𝐹‘𝑤) ∧ (𝐹‘𝑤) ∈ (𝐹‘suc 𝑤)) → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑤)))
29	28	expcomd 420	. . . . . . . . . . . 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 459	. . . . . . . . . 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 725	. . . . . . . . 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 410	. . . . . 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 6867	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (𝐹‘𝑧) = (𝐹‘𝑤))
37	36	eleq1d 2848	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → ((𝐹‘𝑧) ∈ (𝐹‘suc 𝑤) ↔ (𝐹‘𝑤) ∈ (𝐹‘suc 𝑤)))
38	22, 37	syl5ibrcom 249	. . . . . . 7 ⊢ ((∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥) ∧ 𝑤 ∈ ω) → (𝑧 = 𝑤 → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑤)))
39	38	ad4ant23 763	. . . . . 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 870	. . . . 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 244	. . . 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 423	. . 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 7879	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 399   ∨ wo 858   = wceq 1561   ∈ wcel 2143  ∀wral 3077   ⊆ wss 3905  ∅c0 4286  Oncon0 6346  suc csuc 6348  ⟶wf 6517  ‘cfv 6521  ωcom 7846

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pr 5391  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-opab 5164  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-fv 6529  df-om 7847

This theorem is referenced by:  omsmo  8628