Theorem inf3lem6 9540

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9542 for detailed description. (Contributed by NM, 29-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
inf3lem6	⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → 𝐹:ω–1-1→𝒫 𝑥)

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 inf3lem6

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

Step	Hyp	Ref	Expression
1		inf3lem.1	. . . . . . . . . . 11 ⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})
2		inf3lem.2	. . . . . . . . . . 11 ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω)
3		vex 3442	. . . . . . . . . . 11 ⊢ 𝑢 ∈ V
4		vex 3442	. . . . . . . . . . 11 ⊢ 𝑣 ∈ V
5	1, 2, 3, 4	inf3lem5 9539	. . . . . . . . . 10 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ((𝑢 ∈ ω ∧ 𝑣 ∈ 𝑢) → (𝐹‘𝑣) ⊊ (𝐹‘𝑢)))
6		dfpss2 4038	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑣) ⊊ (𝐹‘𝑢) ↔ ((𝐹‘𝑣) ⊆ (𝐹‘𝑢) ∧ ¬ (𝐹‘𝑣) = (𝐹‘𝑢)))
7	6	simprbi 496	. . . . . . . . . 10 ⊢ ((𝐹‘𝑣) ⊊ (𝐹‘𝑢) → ¬ (𝐹‘𝑣) = (𝐹‘𝑢))
8	5, 7	syl6 35	. . . . . . . . 9 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ((𝑢 ∈ ω ∧ 𝑣 ∈ 𝑢) → ¬ (𝐹‘𝑣) = (𝐹‘𝑢)))
9	8	expdimp 452	. . . . . . . 8 ⊢ (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) ∧ 𝑢 ∈ ω) → (𝑣 ∈ 𝑢 → ¬ (𝐹‘𝑣) = (𝐹‘𝑢)))
10	9	adantrl 716	. . . . . . 7 ⊢ (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → (𝑣 ∈ 𝑢 → ¬ (𝐹‘𝑣) = (𝐹‘𝑢)))
11	1, 2, 4, 3	inf3lem5 9539	. . . . . . . . . 10 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ((𝑣 ∈ ω ∧ 𝑢 ∈ 𝑣) → (𝐹‘𝑢) ⊊ (𝐹‘𝑣)))
12		dfpss2 4038	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑢) ⊊ (𝐹‘𝑣) ↔ ((𝐹‘𝑢) ⊆ (𝐹‘𝑣) ∧ ¬ (𝐹‘𝑢) = (𝐹‘𝑣)))
13	12	simprbi 496	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑢) ⊊ (𝐹‘𝑣) → ¬ (𝐹‘𝑢) = (𝐹‘𝑣))
14		eqcom 2741	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ (𝐹‘𝑣) = (𝐹‘𝑢))
15	13, 14	sylnib 328	. . . . . . . . . 10 ⊢ ((𝐹‘𝑢) ⊊ (𝐹‘𝑣) → ¬ (𝐹‘𝑣) = (𝐹‘𝑢))
16	11, 15	syl6 35	. . . . . . . . 9 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ((𝑣 ∈ ω ∧ 𝑢 ∈ 𝑣) → ¬ (𝐹‘𝑣) = (𝐹‘𝑢)))
17	16	expdimp 452	. . . . . . . 8 ⊢ (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) ∧ 𝑣 ∈ ω) → (𝑢 ∈ 𝑣 → ¬ (𝐹‘𝑣) = (𝐹‘𝑢)))
18	17	adantrr 717	. . . . . . 7 ⊢ (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → (𝑢 ∈ 𝑣 → ¬ (𝐹‘𝑣) = (𝐹‘𝑢)))
19	10, 18	jaod 859	. . . . . 6 ⊢ (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → ((𝑣 ∈ 𝑢 ∨ 𝑢 ∈ 𝑣) → ¬ (𝐹‘𝑣) = (𝐹‘𝑢)))
20	19	con2d 134	. . . . 5 ⊢ (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → ((𝐹‘𝑣) = (𝐹‘𝑢) → ¬ (𝑣 ∈ 𝑢 ∨ 𝑢 ∈ 𝑣)))
21		nnord 7814	. . . . . . 7 ⊢ (𝑣 ∈ ω → Ord 𝑣)
22		nnord 7814	. . . . . . 7 ⊢ (𝑢 ∈ ω → Ord 𝑢)
23		ordtri3 6351	. . . . . . 7 ⊢ ((Ord 𝑣 ∧ Ord 𝑢) → (𝑣 = 𝑢 ↔ ¬ (𝑣 ∈ 𝑢 ∨ 𝑢 ∈ 𝑣)))
24	21, 22, 23	syl2an 596	. . . . . 6 ⊢ ((𝑣 ∈ ω ∧ 𝑢 ∈ ω) → (𝑣 = 𝑢 ↔ ¬ (𝑣 ∈ 𝑢 ∨ 𝑢 ∈ 𝑣)))
25	24	adantl 481	. . . . 5 ⊢ (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → (𝑣 = 𝑢 ↔ ¬ (𝑣 ∈ 𝑢 ∨ 𝑢 ∈ 𝑣)))
26	20, 25	sylibrd 259	. . . 4 ⊢ (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → ((𝐹‘𝑣) = (𝐹‘𝑢) → 𝑣 = 𝑢))
27	26	ralrimivva 3177	. . 3 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ∀𝑣 ∈ ω ∀𝑢 ∈ ω ((𝐹‘𝑣) = (𝐹‘𝑢) → 𝑣 = 𝑢))
28		frfnom 8364	. . . . . 6 ⊢ (rec(𝐺, ∅) ↾ ω) Fn ω
29		fneq1 6581	. . . . . 6 ⊢ (𝐹 = (rec(𝐺, ∅) ↾ ω) → (𝐹 Fn ω ↔ (rec(𝐺, ∅) ↾ ω) Fn ω))
30	28, 29	mpbiri 258	. . . . 5 ⊢ (𝐹 = (rec(𝐺, ∅) ↾ ω) → 𝐹 Fn ω)
31		fvelrnb 6892	. . . . . . . 8 ⊢ (𝐹 Fn ω → (𝑢 ∈ ran 𝐹 ↔ ∃𝑣 ∈ ω (𝐹‘𝑣) = 𝑢))
32		inf3lem.4	. . . . . . . . . . . 12 ⊢ 𝐵 ∈ V
33	1, 2, 4, 32	inf3lemd 9534	. . . . . . . . . . 11 ⊢ (𝑣 ∈ ω → (𝐹‘𝑣) ⊆ 𝑥)
34		fvex 6845	. . . . . . . . . . . 12 ⊢ (𝐹‘𝑣) ∈ V
35	34	elpw 4556	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑣) ∈ 𝒫 𝑥 ↔ (𝐹‘𝑣) ⊆ 𝑥)
36	33, 35	sylibr 234	. . . . . . . . . 10 ⊢ (𝑣 ∈ ω → (𝐹‘𝑣) ∈ 𝒫 𝑥)
37		eleq1 2822	. . . . . . . . . 10 ⊢ ((𝐹‘𝑣) = 𝑢 → ((𝐹‘𝑣) ∈ 𝒫 𝑥 ↔ 𝑢 ∈ 𝒫 𝑥))
38	36, 37	syl5ibcom 245	. . . . . . . . 9 ⊢ (𝑣 ∈ ω → ((𝐹‘𝑣) = 𝑢 → 𝑢 ∈ 𝒫 𝑥))
39	38	rexlimiv 3128	. . . . . . . 8 ⊢ (∃𝑣 ∈ ω (𝐹‘𝑣) = 𝑢 → 𝑢 ∈ 𝒫 𝑥)
40	31, 39	biimtrdi 253	. . . . . . 7 ⊢ (𝐹 Fn ω → (𝑢 ∈ ran 𝐹 → 𝑢 ∈ 𝒫 𝑥))
41	40	ssrdv 3937	. . . . . 6 ⊢ (𝐹 Fn ω → ran 𝐹 ⊆ 𝒫 𝑥)
42	41	ancli 548	. . . . 5 ⊢ (𝐹 Fn ω → (𝐹 Fn ω ∧ ran 𝐹 ⊆ 𝒫 𝑥))
43	2, 30, 42	mp2b 10	. . . 4 ⊢ (𝐹 Fn ω ∧ ran 𝐹 ⊆ 𝒫 𝑥)
44		df-f 6494	. . . 4 ⊢ (𝐹:ω⟶𝒫 𝑥 ↔ (𝐹 Fn ω ∧ ran 𝐹 ⊆ 𝒫 𝑥))
45	43, 44	mpbir 231	. . 3 ⊢ 𝐹:ω⟶𝒫 𝑥
46	27, 45	jctil 519	. 2 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹:ω⟶𝒫 𝑥 ∧ ∀𝑣 ∈ ω ∀𝑢 ∈ ω ((𝐹‘𝑣) = (𝐹‘𝑢) → 𝑣 = 𝑢)))
47		dff13 7198	. 2 ⊢ (𝐹:ω–1-1→𝒫 𝑥 ↔ (𝐹:ω⟶𝒫 𝑥 ∧ ∀𝑣 ∈ ω ∀𝑢 ∈ ω ((𝐹‘𝑣) = (𝐹‘𝑢) → 𝑣 = 𝑢)))
48	46, 47	sylibr 234	1 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → 𝐹:ω–1-1→𝒫 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2113   ≠ wne 2930  ∀wral 3049  ∃wrex 3058  {crab 3397  Vcvv 3438   ∩ cin 3898   ⊆ wss 3899   ⊊ wpss 3900  ∅c0 4283  𝒫 cpw 4552  ∪ cuni 4861   ↦ cmpt 5177  ran crn 5623   ↾ cres 5624  Ord word 6314   Fn wfn 6485  ⟶wf 6486  –1-1→wf1 6487  ‘cfv 6490  ωcom 7806  reccrdg 8338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678  ax-reg 9495

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339

This theorem is referenced by:  inf3lem7  9541  dominf  10353  dominfac  10482