Theorem inf3lem1 9665

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9672 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 6906	. . 3 ⊢ (𝑣 = ∅ → (𝐹‘𝑣) = (𝐹‘∅))
2		suceq 6451	. . . 4 ⊢ (𝑣 = ∅ → suc 𝑣 = suc ∅)
3	2	fveq2d 6910	. . 3 ⊢ (𝑣 = ∅ → (𝐹‘suc 𝑣) = (𝐹‘suc ∅))
4	1, 3	sseq12d 4028	. 2 ⊢ (𝑣 = ∅ → ((𝐹‘𝑣) ⊆ (𝐹‘suc 𝑣) ↔ (𝐹‘∅) ⊆ (𝐹‘suc ∅)))
5		fveq2 6906	. . 3 ⊢ (𝑣 = 𝑢 → (𝐹‘𝑣) = (𝐹‘𝑢))
6		suceq 6451	. . . 4 ⊢ (𝑣 = 𝑢 → suc 𝑣 = suc 𝑢)
7	6	fveq2d 6910	. . 3 ⊢ (𝑣 = 𝑢 → (𝐹‘suc 𝑣) = (𝐹‘suc 𝑢))
8	5, 7	sseq12d 4028	. 2 ⊢ (𝑣 = 𝑢 → ((𝐹‘𝑣) ⊆ (𝐹‘suc 𝑣) ↔ (𝐹‘𝑢) ⊆ (𝐹‘suc 𝑢)))
9		fveq2 6906	. . 3 ⊢ (𝑣 = suc 𝑢 → (𝐹‘𝑣) = (𝐹‘suc 𝑢))
10		suceq 6451	. . . 4 ⊢ (𝑣 = suc 𝑢 → suc 𝑣 = suc suc 𝑢)
11	10	fveq2d 6910	. . 3 ⊢ (𝑣 = suc 𝑢 → (𝐹‘suc 𝑣) = (𝐹‘suc suc 𝑢))
12	9, 11	sseq12d 4028	. 2 ⊢ (𝑣 = suc 𝑢 → ((𝐹‘𝑣) ⊆ (𝐹‘suc 𝑣) ↔ (𝐹‘suc 𝑢) ⊆ (𝐹‘suc suc 𝑢)))
13		fveq2 6906	. . 3 ⊢ (𝑣 = 𝐴 → (𝐹‘𝑣) = (𝐹‘𝐴))
14		suceq 6451	. . . 4 ⊢ (𝑣 = 𝐴 → suc 𝑣 = suc 𝐴)
15	14	fveq2d 6910	. . 3 ⊢ (𝑣 = 𝐴 → (𝐹‘suc 𝑣) = (𝐹‘suc 𝐴))
16	13, 15	sseq12d 4028	. 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 9662	. . 3 ⊢ (𝐹‘∅) = ∅
21		0ss 4405	. . 3 ⊢ ∅ ⊆ (𝐹‘suc ∅)
22	20, 21	eqsstri 4029	. 2 ⊢ (𝐹‘∅) ⊆ (𝐹‘suc ∅)
23		sstr2 4001	. . . . . . . 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 612	. . . . . 6 ⊢ ((𝐹‘𝑢) ⊆ (𝐹‘suc 𝑢) → ((𝑣 ∈ 𝑥 ∧ (𝑣 ∩ 𝑥) ⊆ (𝐹‘𝑢)) → (𝑣 ∈ 𝑥 ∧ (𝑣 ∩ 𝑥) ⊆ (𝐹‘suc 𝑢))))
26		vex 3481	. . . . . . . . . 10 ⊢ 𝑢 ∈ V
27	17, 18, 26, 19	inf3lemc 9663	. . . . . . . . 9 ⊢ (𝑢 ∈ ω → (𝐹‘suc 𝑢) = (𝐺‘(𝐹‘𝑢)))
28	27	eleq2d 2824	. . . . . . . 8 ⊢ (𝑢 ∈ ω → (𝑣 ∈ (𝐹‘suc 𝑢) ↔ 𝑣 ∈ (𝐺‘(𝐹‘𝑢))))
29		vex 3481	. . . . . . . . 9 ⊢ 𝑣 ∈ V
30		fvex 6919	. . . . . . . . 9 ⊢ (𝐹‘𝑢) ∈ V
31	17, 18, 29, 30	inf3lema 9661	. . . . . . . 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 7903	. . . . . . . . . 10 ⊢ (𝑢 ∈ ω ↔ suc 𝑢 ∈ ω)
34	26	sucex 7825	. . . . . . . . . . 11 ⊢ suc 𝑢 ∈ V
35	17, 18, 34, 19	inf3lemc 9663	. . . . . . . . . 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 2824	. . . . . . . 8 ⊢ (𝑢 ∈ ω → (𝑣 ∈ (𝐹‘suc suc 𝑢) ↔ 𝑣 ∈ (𝐺‘(𝐹‘suc 𝑢))))
38		fvex 6919	. . . . . . . . 9 ⊢ (𝐹‘suc 𝑢) ∈ V
39	17, 18, 29, 38	inf3lema 9661	. . . . . . . 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 4000	. . 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 7918	1 ⊢ (𝐴 ∈ ω → (𝐹‘𝐴) ⊆ (𝐹‘suc 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1536   ∈ wcel 2105  {crab 3432  Vcvv 3477   ∩ cin 3961   ⊆ wss 3962  ∅c0 4338   ↦ cmpt 5230   ↾ cres 5690  suc csuc 6387  ‘cfv 6562  ωcom 7886  reccrdg 8447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448

This theorem is referenced by:  inf3lem4  9668