Theorem inf3lem1 9593

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9600 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 6881	. . 3 ⊢ (𝑣 = ∅ → (𝐹‘𝑣) = (𝐹‘∅))
2		suceq 6429	. . . 4 ⊢ (𝑣 = ∅ → suc 𝑣 = suc ∅)
3	2	fveq2d 6885	. . 3 ⊢ (𝑣 = ∅ → (𝐹‘suc 𝑣) = (𝐹‘suc ∅))
4	1, 3	sseq12d 3970	. 2 ⊢ (𝑣 = ∅ → ((𝐹‘𝑣) ⊆ (𝐹‘suc 𝑣) ↔ (𝐹‘∅) ⊆ (𝐹‘suc ∅)))
5		fveq2 6881	. . 3 ⊢ (𝑣 = 𝑢 → (𝐹‘𝑣) = (𝐹‘𝑢))
6		suceq 6429	. . . 4 ⊢ (𝑣 = 𝑢 → suc 𝑣 = suc 𝑢)
7	6	fveq2d 6885	. . 3 ⊢ (𝑣 = 𝑢 → (𝐹‘suc 𝑣) = (𝐹‘suc 𝑢))
8	5, 7	sseq12d 3970	. 2 ⊢ (𝑣 = 𝑢 → ((𝐹‘𝑣) ⊆ (𝐹‘suc 𝑣) ↔ (𝐹‘𝑢) ⊆ (𝐹‘suc 𝑢)))
9		fveq2 6881	. . 3 ⊢ (𝑣 = suc 𝑢 → (𝐹‘𝑣) = (𝐹‘suc 𝑢))
10		suceq 6429	. . . 4 ⊢ (𝑣 = suc 𝑢 → suc 𝑣 = suc suc 𝑢)
11	10	fveq2d 6885	. . 3 ⊢ (𝑣 = suc 𝑢 → (𝐹‘suc 𝑣) = (𝐹‘suc suc 𝑢))
12	9, 11	sseq12d 3970	. 2 ⊢ (𝑣 = suc 𝑢 → ((𝐹‘𝑣) ⊆ (𝐹‘suc 𝑣) ↔ (𝐹‘suc 𝑢) ⊆ (𝐹‘suc suc 𝑢)))
13		fveq2 6881	. . 3 ⊢ (𝑣 = 𝐴 → (𝐹‘𝑣) = (𝐹‘𝐴))
14		suceq 6429	. . . 4 ⊢ (𝑣 = 𝐴 → suc 𝑣 = suc 𝐴)
15	14	fveq2d 6885	. . 3 ⊢ (𝑣 = 𝐴 → (𝐹‘suc 𝑣) = (𝐹‘suc 𝐴))
16	13, 15	sseq12d 3970	. 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 9590	. . 3 ⊢ (𝐹‘∅) = ∅
21		0ss 4357	. . 3 ⊢ ∅ ⊆ (𝐹‘suc ∅)
22	20, 21	eqsstri 3983	. 2 ⊢ (𝐹‘∅) ⊆ (𝐹‘suc ∅)
23		sstr2 3944	. . . . . . . 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 623	. . . . . 6 ⊢ ((𝐹‘𝑢) ⊆ (𝐹‘suc 𝑢) → ((𝑣 ∈ 𝑥 ∧ (𝑣 ∩ 𝑥) ⊆ (𝐹‘𝑢)) → (𝑣 ∈ 𝑥 ∧ (𝑣 ∩ 𝑥) ⊆ (𝐹‘suc 𝑢))))
26		vex 3459	. . . . . . . . . 10 ⊢ 𝑢 ∈ V
27	17, 18, 26, 19	inf3lemc 9591	. . . . . . . . 9 ⊢ (𝑢 ∈ ω → (𝐹‘suc 𝑢) = (𝐺‘(𝐹‘𝑢)))
28	27	eleq2d 2849	. . . . . . . 8 ⊢ (𝑢 ∈ ω → (𝑣 ∈ (𝐹‘suc 𝑢) ↔ 𝑣 ∈ (𝐺‘(𝐹‘𝑢))))
29		vex 3459	. . . . . . . . 9 ⊢ 𝑣 ∈ V
30		fvex 6894	. . . . . . . . 9 ⊢ (𝐹‘𝑢) ∈ V
31	17, 18, 29, 30	inf3lema 9589	. . . . . . . 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 7875	. . . . . . . . . 10 ⊢ (𝑢 ∈ ω ↔ suc 𝑢 ∈ ω)
34	26	sucex 7801	. . . . . . . . . . 11 ⊢ suc 𝑢 ∈ V
35	17, 18, 34, 19	inf3lemc 9591	. . . . . . . . . 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 2849	. . . . . . . 8 ⊢ (𝑢 ∈ ω → (𝑣 ∈ (𝐹‘suc suc 𝑢) ↔ 𝑣 ∈ (𝐺‘(𝐹‘suc 𝑢))))
38		fvex 6894	. . . . . . . . 9 ⊢ (𝐹‘suc 𝑢) ∈ V
39	17, 18, 29, 38	inf3lema 9589	. . . . . . . 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 411	. . . 4 ⊢ ((𝑢 ∈ ω ∧ (𝐹‘𝑢) ⊆ (𝐹‘suc 𝑢)) → (𝑣 ∈ (𝐹‘suc 𝑢) → 𝑣 ∈ (𝐹‘suc suc 𝑢)))
44	43	ssrdv 3943	. . 3 ⊢ ((𝑢 ∈ ω ∧ (𝐹‘𝑢) ⊆ (𝐹‘suc 𝑢)) → (𝐹‘suc 𝑢) ⊆ (𝐹‘suc suc 𝑢))
45	44	ex 417	. 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))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {crab 3416  Vcvv 3455   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286   ↦ cmpt 5192   ↾ cres 5663  suc csuc 6362  ‘cfv 6536  ωcom 7858  reccrdg 8392

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-om 7859  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393

This theorem is used by:  inf3lem4  9596