Theorem inf3lem1 9697

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9704 for detailed description. (Contributed by NM, 28-Oct-1996.)

Hypotheses

Ref	Expression
inf3lem.1	⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})
inf3lem.2	⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω)
inf3lem.3	⊢ 𝐴 ∈ V
inf3lem.4	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
inf3lem1	⊢ (𝐴 ∈ ω → (𝐹‘𝐴) ⊆ (𝐹‘suc 𝐴))

Distinct variable group:   𝑥,𝑦,𝑤

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑤)   𝐵(𝑥,𝑦,𝑤)   𝐹(𝑥,𝑦,𝑤)   𝐺(𝑥,𝑦,𝑤)

Proof of Theorem inf3lem1

Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6920	. . 3 ⊢ (𝑣 = ∅ → (𝐹‘𝑣) = (𝐹‘∅))
2		suceq 6461	. . . 4 ⊢ (𝑣 = ∅ → suc 𝑣 = suc ∅)
3	2	fveq2d 6924	. . 3 ⊢ (𝑣 = ∅ → (𝐹‘suc 𝑣) = (𝐹‘suc ∅))
4	1, 3	sseq12d 4042	. 2 ⊢ (𝑣 = ∅ → ((𝐹‘𝑣) ⊆ (𝐹‘suc 𝑣) ↔ (𝐹‘∅) ⊆ (𝐹‘suc ∅)))
5		fveq2 6920	. . 3 ⊢ (𝑣 = 𝑢 → (𝐹‘𝑣) = (𝐹‘𝑢))
6		suceq 6461	. . . 4 ⊢ (𝑣 = 𝑢 → suc 𝑣 = suc 𝑢)
7	6	fveq2d 6924	. . 3 ⊢ (𝑣 = 𝑢 → (𝐹‘suc 𝑣) = (𝐹‘suc 𝑢))
8	5, 7	sseq12d 4042	. 2 ⊢ (𝑣 = 𝑢 → ((𝐹‘𝑣) ⊆ (𝐹‘suc 𝑣) ↔ (𝐹‘𝑢) ⊆ (𝐹‘suc 𝑢)))
9		fveq2 6920	. . 3 ⊢ (𝑣 = suc 𝑢 → (𝐹‘𝑣) = (𝐹‘suc 𝑢))
10		suceq 6461	. . . 4 ⊢ (𝑣 = suc 𝑢 → suc 𝑣 = suc suc 𝑢)
11	10	fveq2d 6924	. . 3 ⊢ (𝑣 = suc 𝑢 → (𝐹‘suc 𝑣) = (𝐹‘suc suc 𝑢))
12	9, 11	sseq12d 4042	. 2 ⊢ (𝑣 = suc 𝑢 → ((𝐹‘𝑣) ⊆ (𝐹‘suc 𝑣) ↔ (𝐹‘suc 𝑢) ⊆ (𝐹‘suc suc 𝑢)))
13		fveq2 6920	. . 3 ⊢ (𝑣 = 𝐴 → (𝐹‘𝑣) = (𝐹‘𝐴))
14		suceq 6461	. . . 4 ⊢ (𝑣 = 𝐴 → suc 𝑣 = suc 𝐴)
15	14	fveq2d 6924	. . 3 ⊢ (𝑣 = 𝐴 → (𝐹‘suc 𝑣) = (𝐹‘suc 𝐴))
16	13, 15	sseq12d 4042	. 2 ⊢ (𝑣 = 𝐴 → ((𝐹‘𝑣) ⊆ (𝐹‘suc 𝑣) ↔ (𝐹‘𝐴) ⊆ (𝐹‘suc 𝐴)))
17		inf3lem.1	. . . 4 ⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})
18		inf3lem.2	. . . 4 ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω)
19		inf3lem.3	. . . 4 ⊢ 𝐴 ∈ V
20	17, 18, 19, 19	inf3lemb 9694	. . 3 ⊢ (𝐹‘∅) = ∅
21		0ss 4423	. . 3 ⊢ ∅ ⊆ (𝐹‘suc ∅)
22	20, 21	eqsstri 4043	. 2 ⊢ (𝐹‘∅) ⊆ (𝐹‘suc ∅)
23		sstr2 4015	. . . . . . . 8 ⊢ ((𝑣 ∩ 𝑥) ⊆ (𝐹‘𝑢) → ((𝐹‘𝑢) ⊆ (𝐹‘suc 𝑢) → (𝑣 ∩ 𝑥) ⊆ (𝐹‘suc 𝑢)))
24	23	com12 32	. . . . . . 7 ⊢ ((𝐹‘𝑢) ⊆ (𝐹‘suc 𝑢) → ((𝑣 ∩ 𝑥) ⊆ (𝐹‘𝑢) → (𝑣 ∩ 𝑥) ⊆ (𝐹‘suc 𝑢)))
25	24	anim2d 611	. . . . . 6 ⊢ ((𝐹‘𝑢) ⊆ (𝐹‘suc 𝑢) → ((𝑣 ∈ 𝑥 ∧ (𝑣 ∩ 𝑥) ⊆ (𝐹‘𝑢)) → (𝑣 ∈ 𝑥 ∧ (𝑣 ∩ 𝑥) ⊆ (𝐹‘suc 𝑢))))
26		vex 3492	. . . . . . . . . 10 ⊢ 𝑢 ∈ V
27	17, 18, 26, 19	inf3lemc 9695	. . . . . . . . 9 ⊢ (𝑢 ∈ ω → (𝐹‘suc 𝑢) = (𝐺‘(𝐹‘𝑢)))
28	27	eleq2d 2830	. . . . . . . 8 ⊢ (𝑢 ∈ ω → (𝑣 ∈ (𝐹‘suc 𝑢) ↔ 𝑣 ∈ (𝐺‘(𝐹‘𝑢))))
29		vex 3492	. . . . . . . . 9 ⊢ 𝑣 ∈ V
30		fvex 6933	. . . . . . . . 9 ⊢ (𝐹‘𝑢) ∈ V
31	17, 18, 29, 30	inf3lema 9693	. . . . . . . 8 ⊢ (𝑣 ∈ (𝐺‘(𝐹‘𝑢)) ↔ (𝑣 ∈ 𝑥 ∧ (𝑣 ∩ 𝑥) ⊆ (𝐹‘𝑢)))
32	28, 31	bitrdi 287	. . . . . . 7 ⊢ (𝑢 ∈ ω → (𝑣 ∈ (𝐹‘suc 𝑢) ↔ (𝑣 ∈ 𝑥 ∧ (𝑣 ∩ 𝑥) ⊆ (𝐹‘𝑢))))
33		peano2b 7920	. . . . . . . . . 10 ⊢ (𝑢 ∈ ω ↔ suc 𝑢 ∈ ω)
34	26	sucex 7842	. . . . . . . . . . 11 ⊢ suc 𝑢 ∈ V
35	17, 18, 34, 19	inf3lemc 9695	. . . . . . . . . 10 ⊢ (suc 𝑢 ∈ ω → (𝐹‘suc suc 𝑢) = (𝐺‘(𝐹‘suc 𝑢)))
36	33, 35	sylbi 217	. . . . . . . . 9 ⊢ (𝑢 ∈ ω → (𝐹‘suc suc 𝑢) = (𝐺‘(𝐹‘suc 𝑢)))
37	36	eleq2d 2830	. . . . . . . 8 ⊢ (𝑢 ∈ ω → (𝑣 ∈ (𝐹‘suc suc 𝑢) ↔ 𝑣 ∈ (𝐺‘(𝐹‘suc 𝑢))))
38		fvex 6933	. . . . . . . . 9 ⊢ (𝐹‘suc 𝑢) ∈ V
39	17, 18, 29, 38	inf3lema 9693	. . . . . . . 8 ⊢ (𝑣 ∈ (𝐺‘(𝐹‘suc 𝑢)) ↔ (𝑣 ∈ 𝑥 ∧ (𝑣 ∩ 𝑥) ⊆ (𝐹‘suc 𝑢)))
40	37, 39	bitrdi 287	. . . . . . 7 ⊢ (𝑢 ∈ ω → (𝑣 ∈ (𝐹‘suc suc 𝑢) ↔ (𝑣 ∈ 𝑥 ∧ (𝑣 ∩ 𝑥) ⊆ (𝐹‘suc 𝑢))))
41	32, 40	imbi12d 344	. . . . . 6 ⊢ (𝑢 ∈ ω → ((𝑣 ∈ (𝐹‘suc 𝑢) → 𝑣 ∈ (𝐹‘suc suc 𝑢)) ↔ ((𝑣 ∈ 𝑥 ∧ (𝑣 ∩ 𝑥) ⊆ (𝐹‘𝑢)) → (𝑣 ∈ 𝑥 ∧ (𝑣 ∩ 𝑥) ⊆ (𝐹‘suc 𝑢)))))
42	25, 41	imbitrrid 246	. . . . 5 ⊢ (𝑢 ∈ ω → ((𝐹‘𝑢) ⊆ (𝐹‘suc 𝑢) → (𝑣 ∈ (𝐹‘suc 𝑢) → 𝑣 ∈ (𝐹‘suc suc 𝑢))))
43	42	imp 406	. . . 4 ⊢ ((𝑢 ∈ ω ∧ (𝐹‘𝑢) ⊆ (𝐹‘suc 𝑢)) → (𝑣 ∈ (𝐹‘suc 𝑢) → 𝑣 ∈ (𝐹‘suc suc 𝑢)))
44	43	ssrdv 4014	. . 3 ⊢ ((𝑢 ∈ ω ∧ (𝐹‘𝑢) ⊆ (𝐹‘suc 𝑢)) → (𝐹‘suc 𝑢) ⊆ (𝐹‘suc suc 𝑢))
45	44	ex 412	. 2 ⊢ (𝑢 ∈ ω → ((𝐹‘𝑢) ⊆ (𝐹‘suc 𝑢) → (𝐹‘suc 𝑢) ⊆ (𝐹‘suc suc 𝑢)))
46	4, 8, 12, 16, 22, 45	finds 7936	1 ⊢ (𝐴 ∈ ω → (𝐹‘𝐴) ⊆ (𝐹‘suc 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {crab 3443  Vcvv 3488   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352   ↦ cmpt 5249   ↾ cres 5702  suc csuc 6397  ‘cfv 6573  ωcom 7903  reccrdg 8465

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466

This theorem is referenced by:  inf3lem4  9700