Theorem inf3lem1 9623

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9630 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 6892	. . 3 ⊢ (𝑣 = ∅ → (𝐹‘𝑣) = (𝐹‘∅))
2		suceq 6431	. . . 4 ⊢ (𝑣 = ∅ → suc 𝑣 = suc ∅)
3	2	fveq2d 6896	. . 3 ⊢ (𝑣 = ∅ → (𝐹‘suc 𝑣) = (𝐹‘suc ∅))
4	1, 3	sseq12d 4016	. 2 ⊢ (𝑣 = ∅ → ((𝐹‘𝑣) ⊆ (𝐹‘suc 𝑣) ↔ (𝐹‘∅) ⊆ (𝐹‘suc ∅)))
5		fveq2 6892	. . 3 ⊢ (𝑣 = 𝑢 → (𝐹‘𝑣) = (𝐹‘𝑢))
6		suceq 6431	. . . 4 ⊢ (𝑣 = 𝑢 → suc 𝑣 = suc 𝑢)
7	6	fveq2d 6896	. . 3 ⊢ (𝑣 = 𝑢 → (𝐹‘suc 𝑣) = (𝐹‘suc 𝑢))
8	5, 7	sseq12d 4016	. 2 ⊢ (𝑣 = 𝑢 → ((𝐹‘𝑣) ⊆ (𝐹‘suc 𝑣) ↔ (𝐹‘𝑢) ⊆ (𝐹‘suc 𝑢)))
9		fveq2 6892	. . 3 ⊢ (𝑣 = suc 𝑢 → (𝐹‘𝑣) = (𝐹‘suc 𝑢))
10		suceq 6431	. . . 4 ⊢ (𝑣 = suc 𝑢 → suc 𝑣 = suc suc 𝑢)
11	10	fveq2d 6896	. . 3 ⊢ (𝑣 = suc 𝑢 → (𝐹‘suc 𝑣) = (𝐹‘suc suc 𝑢))
12	9, 11	sseq12d 4016	. 2 ⊢ (𝑣 = suc 𝑢 → ((𝐹‘𝑣) ⊆ (𝐹‘suc 𝑣) ↔ (𝐹‘suc 𝑢) ⊆ (𝐹‘suc suc 𝑢)))
13		fveq2 6892	. . 3 ⊢ (𝑣 = 𝐴 → (𝐹‘𝑣) = (𝐹‘𝐴))
14		suceq 6431	. . . 4 ⊢ (𝑣 = 𝐴 → suc 𝑣 = suc 𝐴)
15	14	fveq2d 6896	. . 3 ⊢ (𝑣 = 𝐴 → (𝐹‘suc 𝑣) = (𝐹‘suc 𝐴))
16	13, 15	sseq12d 4016	. 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 9620	. . 3 ⊢ (𝐹‘∅) = ∅
21		0ss 4397	. . 3 ⊢ ∅ ⊆ (𝐹‘suc ∅)
22	20, 21	eqsstri 4017	. 2 ⊢ (𝐹‘∅) ⊆ (𝐹‘suc ∅)
23		sstr2 3990	. . . . . . . 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 613	. . . . . 6 ⊢ ((𝐹‘𝑢) ⊆ (𝐹‘suc 𝑢) → ((𝑣 ∈ 𝑥 ∧ (𝑣 ∩ 𝑥) ⊆ (𝐹‘𝑢)) → (𝑣 ∈ 𝑥 ∧ (𝑣 ∩ 𝑥) ⊆ (𝐹‘suc 𝑢))))
26		vex 3479	. . . . . . . . . 10 ⊢ 𝑢 ∈ V
27	17, 18, 26, 19	inf3lemc 9621	. . . . . . . . 9 ⊢ (𝑢 ∈ ω → (𝐹‘suc 𝑢) = (𝐺‘(𝐹‘𝑢)))
28	27	eleq2d 2820	. . . . . . . 8 ⊢ (𝑢 ∈ ω → (𝑣 ∈ (𝐹‘suc 𝑢) ↔ 𝑣 ∈ (𝐺‘(𝐹‘𝑢))))
29		vex 3479	. . . . . . . . 9 ⊢ 𝑣 ∈ V
30		fvex 6905	. . . . . . . . 9 ⊢ (𝐹‘𝑢) ∈ V
31	17, 18, 29, 30	inf3lema 9619	. . . . . . . 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 7872	. . . . . . . . . 10 ⊢ (𝑢 ∈ ω ↔ suc 𝑢 ∈ ω)
34	26	sucex 7794	. . . . . . . . . . 11 ⊢ suc 𝑢 ∈ V
35	17, 18, 34, 19	inf3lemc 9621	. . . . . . . . . 10 ⊢ (suc 𝑢 ∈ ω → (𝐹‘suc suc 𝑢) = (𝐺‘(𝐹‘suc 𝑢)))
36	33, 35	sylbi 216	. . . . . . . . 9 ⊢ (𝑢 ∈ ω → (𝐹‘suc suc 𝑢) = (𝐺‘(𝐹‘suc 𝑢)))
37	36	eleq2d 2820	. . . . . . . 8 ⊢ (𝑢 ∈ ω → (𝑣 ∈ (𝐹‘suc suc 𝑢) ↔ 𝑣 ∈ (𝐺‘(𝐹‘suc 𝑢))))
38		fvex 6905	. . . . . . . . 9 ⊢ (𝐹‘suc 𝑢) ∈ V
39	17, 18, 29, 38	inf3lema 9619	. . . . . . . 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 345	. . . . . 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 245	. . . . 5 ⊢ (𝑢 ∈ ω → ((𝐹‘𝑢) ⊆ (𝐹‘suc 𝑢) → (𝑣 ∈ (𝐹‘suc 𝑢) → 𝑣 ∈ (𝐹‘suc suc 𝑢))))
43	42	imp 408	. . . 4 ⊢ ((𝑢 ∈ ω ∧ (𝐹‘𝑢) ⊆ (𝐹‘suc 𝑢)) → (𝑣 ∈ (𝐹‘suc 𝑢) → 𝑣 ∈ (𝐹‘suc suc 𝑢)))
44	43	ssrdv 3989	. . 3 ⊢ ((𝑢 ∈ ω ∧ (𝐹‘𝑢) ⊆ (𝐹‘suc 𝑢)) → (𝐹‘suc 𝑢) ⊆ (𝐹‘suc suc 𝑢))
45	44	ex 414	. 2 ⊢ (𝑢 ∈ ω → ((𝐹‘𝑢) ⊆ (𝐹‘suc 𝑢) → (𝐹‘suc 𝑢) ⊆ (𝐹‘suc suc 𝑢)))
46	4, 8, 12, 16, 22, 45	finds 7889	1 ⊢ (𝐴 ∈ ω → (𝐹‘𝐴) ⊆ (𝐹‘suc 𝐴))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {crab 3433  Vcvv 3475   ∩ cin 3948   ⊆ wss 3949  ∅c0 4323   ↦ cmpt 5232   ↾ cres 5679  suc csuc 6367  ‘cfv 6544  ωcom 7855  reccrdg 8409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410

This theorem is referenced by:  inf3lem4  9626