Theorem inf3lem1 9653

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9660 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 6905	. . 3 ⊢ (𝑣 = ∅ → (𝐹‘𝑣) = (𝐹‘∅))
2		suceq 6446	. . . 4 ⊢ (𝑣 = ∅ → suc 𝑣 = suc ∅)
3	2	fveq2d 6909	. . 3 ⊢ (𝑣 = ∅ → (𝐹‘suc 𝑣) = (𝐹‘suc ∅))
4	1, 3	sseq12d 3946	. 2 ⊢ (𝑣 = ∅ → ((𝐹‘𝑣) ⊆ (𝐹‘suc 𝑣) ↔ (𝐹‘∅) ⊆ (𝐹‘suc ∅)))
5		fveq2 6905	. . 3 ⊢ (𝑣 = 𝑢 → (𝐹‘𝑣) = (𝐹‘𝑢))
6		suceq 6446	. . . 4 ⊢ (𝑣 = 𝑢 → suc 𝑣 = suc 𝑢)
7	6	fveq2d 6909	. . 3 ⊢ (𝑣 = 𝑢 → (𝐹‘suc 𝑣) = (𝐹‘suc 𝑢))
8	5, 7	sseq12d 3946	. 2 ⊢ (𝑣 = 𝑢 → ((𝐹‘𝑣) ⊆ (𝐹‘suc 𝑣) ↔ (𝐹‘𝑢) ⊆ (𝐹‘suc 𝑢)))
9		fveq2 6905	. . 3 ⊢ (𝑣 = suc 𝑢 → (𝐹‘𝑣) = (𝐹‘suc 𝑢))
10		suceq 6446	. . . 4 ⊢ (𝑣 = suc 𝑢 → suc 𝑣 = suc suc 𝑢)
11	10	fveq2d 6909	. . 3 ⊢ (𝑣 = suc 𝑢 → (𝐹‘suc 𝑣) = (𝐹‘suc suc 𝑢))
12	9, 11	sseq12d 3946	. 2 ⊢ (𝑣 = suc 𝑢 → ((𝐹‘𝑣) ⊆ (𝐹‘suc 𝑣) ↔ (𝐹‘suc 𝑢) ⊆ (𝐹‘suc suc 𝑢)))
13		fveq2 6905	. . 3 ⊢ (𝑣 = 𝐴 → (𝐹‘𝑣) = (𝐹‘𝐴))
14		suceq 6446	. . . 4 ⊢ (𝑣 = 𝐴 → suc 𝑣 = suc 𝐴)
15	14	fveq2d 6909	. . 3 ⊢ (𝑣 = 𝐴 → (𝐹‘suc 𝑣) = (𝐹‘suc 𝐴))
16	13, 15	sseq12d 3946	. 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 9650	. . 3 ⊢ (𝐹‘∅) = ∅
21		0ss 4353	. . 3 ⊢ ∅ ⊆ (𝐹‘suc ∅)
22	20, 21	eqsstri 3959	. 2 ⊢ (𝐹‘∅) ⊆ (𝐹‘suc ∅)
23		sstr2 3920	. . . . . . . 8 ⊢ ((𝑣 ∩ 𝑥) ⊆ (𝐹‘𝑢) → ((𝐹‘𝑢) ⊆ (𝐹‘suc 𝑢) → (𝑣 ∩ 𝑥) ⊆ (𝐹‘suc 𝑢)))
24	23	com12 33	. . . . . . 7 ⊢ ((𝐹‘𝑢) ⊆ (𝐹‘suc 𝑢) → ((𝑣 ∩ 𝑥) ⊆ (𝐹‘𝑢) → (𝑣 ∩ 𝑥) ⊆ (𝐹‘suc 𝑢)))
25	24	anim2d 624	. . . . . 6 ⊢ ((𝐹‘𝑢) ⊆ (𝐹‘suc 𝑢) → ((𝑣 ∈ 𝑥 ∧ (𝑣 ∩ 𝑥) ⊆ (𝐹‘𝑢)) → (𝑣 ∈ 𝑥 ∧ (𝑣 ∩ 𝑥) ⊆ (𝐹‘suc 𝑢))))
26		vex 3458	. . . . . . . . . 10 ⊢ 𝑢 ∈ V
27	17, 18, 26, 19	inf3lemc 9651	. . . . . . . . 9 ⊢ (𝑢 ∈ ω → (𝐹‘suc 𝑢) = (𝐺‘(𝐹‘𝑢)))
28	27	eleq2d 2850	. . . . . . . 8 ⊢ (𝑢 ∈ ω → (𝑣 ∈ (𝐹‘suc 𝑢) ↔ 𝑣 ∈ (𝐺‘(𝐹‘𝑢))))
29		vex 3458	. . . . . . . . 9 ⊢ 𝑣 ∈ V
30		fvex 6918	. . . . . . . . 9 ⊢ (𝐹‘𝑢) ∈ V
31	17, 18, 29, 30	inf3lema 9649	. . . . . . . 8 ⊢ (𝑣 ∈ (𝐺‘(𝐹‘𝑢)) ↔ (𝑣 ∈ 𝑥 ∧ (𝑣 ∩ 𝑥) ⊆ (𝐹‘𝑢)))
32	28, 31	bitrdi 290	. . . . . . 7 ⊢ (𝑢 ∈ ω → (𝑣 ∈ (𝐹‘suc 𝑢) ↔ (𝑣 ∈ 𝑥 ∧ (𝑣 ∩ 𝑥) ⊆ (𝐹‘𝑢))))
33		peano2b 7915	. . . . . . . . . 10 ⊢ (𝑢 ∈ ω ↔ suc 𝑢 ∈ ω)
34	26	sucex 7841	. . . . . . . . . . 11 ⊢ suc 𝑢 ∈ V
35	17, 18, 34, 19	inf3lemc 9651	. . . . . . . . . 10 ⊢ (suc 𝑢 ∈ ω → (𝐹‘suc suc 𝑢) = (𝐺‘(𝐹‘suc 𝑢)))
36	33, 35	sylbi 220	. . . . . . . . 9 ⊢ (𝑢 ∈ ω → (𝐹‘suc suc 𝑢) = (𝐺‘(𝐹‘suc 𝑢)))
37	36	eleq2d 2850	. . . . . . . 8 ⊢ (𝑢 ∈ ω → (𝑣 ∈ (𝐹‘suc suc 𝑢) ↔ 𝑣 ∈ (𝐺‘(𝐹‘suc 𝑢))))
38		fvex 6918	. . . . . . . . 9 ⊢ (𝐹‘suc 𝑢) ∈ V
39	17, 18, 29, 38	inf3lema 9649	. . . . . . . 8 ⊢ (𝑣 ∈ (𝐺‘(𝐹‘suc 𝑢)) ↔ (𝑣 ∈ 𝑥 ∧ (𝑣 ∩ 𝑥) ⊆ (𝐹‘suc 𝑢)))
40	37, 39	bitrdi 290	. . . . . . 7 ⊢ (𝑢 ∈ ω → (𝑣 ∈ (𝐹‘suc suc 𝑢) ↔ (𝑣 ∈ 𝑥 ∧ (𝑣 ∩ 𝑥) ⊆ (𝐹‘suc 𝑢))))
41	32, 40	imbi12d 347	. . . . . 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 249	. . . . 5 ⊢ (𝑢 ∈ ω → ((𝐹‘𝑢) ⊆ (𝐹‘suc 𝑢) → (𝑣 ∈ (𝐹‘suc 𝑢) → 𝑣 ∈ (𝐹‘suc suc 𝑢))))
43	42	imp 412	. . . 4 ⊢ ((𝑢 ∈ ω ∧ (𝐹‘𝑢) ⊆ (𝐹‘suc 𝑢)) → (𝑣 ∈ (𝐹‘suc 𝑢) → 𝑣 ∈ (𝐹‘suc suc 𝑢)))
44	43	ssrdv 3919	. . 3 ⊢ ((𝑢 ∈ ω ∧ (𝐹‘𝑢) ⊆ (𝐹‘suc 𝑢)) → (𝐹‘suc 𝑢) ⊆ (𝐹‘suc suc 𝑢))
45	44	ex 418	. 2 ⊢ (𝑢 ∈ ω → ((𝐹‘𝑢) ⊆ (𝐹‘suc 𝑢) → (𝐹‘suc 𝑢) ⊆ (𝐹‘suc suc 𝑢)))
46	4, 8, 12, 16, 22, 45	finds 7929	1 ⊢ (𝐴 ∈ ω → (𝐹‘𝐴) ⊆ (𝐹‘suc 𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {crab 3416  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  ∅c0 4281   ↦ cmpt 5189   ↾ cres 5657  suc csuc 6379  ‘cfv 6555  ωcom 7898  reccrdg 8442

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7772

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ov 7443  df-om 7899  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443

This theorem is used by:  inf3lem4  9656