Theorem inf3lem6 9548

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9550 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 3434	. . . . . . . . . . 11 ⊢ 𝑢 ∈ V
4		vex 3434	. . . . . . . . . . 11 ⊢ 𝑣 ∈ V
5	1, 2, 3, 4	inf3lem5 9547	. . . . . . . . . 10 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ((𝑢 ∈ ω ∧ 𝑣 ∈ 𝑢) → (𝐹‘𝑣) ⊊ (𝐹‘𝑢)))
6		dfpss2 4029	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑣) ⊊ (𝐹‘𝑢) ↔ ((𝐹‘𝑣) ⊆ (𝐹‘𝑢) ∧ ¬ (𝐹‘𝑣) = (𝐹‘𝑢)))
7	6	simprbi 497	. . . . . . . . . 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 717	. . . . . . 7 ⊢ (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → (𝑣 ∈ 𝑢 → ¬ (𝐹‘𝑣) = (𝐹‘𝑢)))
11	1, 2, 4, 3	inf3lem5 9547	. . . . . . . . . 10 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ((𝑣 ∈ ω ∧ 𝑢 ∈ 𝑣) → (𝐹‘𝑢) ⊊ (𝐹‘𝑣)))
12		dfpss2 4029	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑢) ⊊ (𝐹‘𝑣) ↔ ((𝐹‘𝑢) ⊆ (𝐹‘𝑣) ∧ ¬ (𝐹‘𝑢) = (𝐹‘𝑣)))
13	12	simprbi 497	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑢) ⊊ (𝐹‘𝑣) → ¬ (𝐹‘𝑢) = (𝐹‘𝑣))
14		eqcom 2744	. . . . . . . . . . 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 718	. . . . . . 7 ⊢ (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → (𝑢 ∈ 𝑣 → ¬ (𝐹‘𝑣) = (𝐹‘𝑢)))
19	10, 18	jaod 860	. . . . . 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 7819	. . . . . . 7 ⊢ (𝑣 ∈ ω → Ord 𝑣)
22		nnord 7819	. . . . . . 7 ⊢ (𝑢 ∈ ω → Ord 𝑢)
23		ordtri3 6354	. . . . . . 7 ⊢ ((Ord 𝑣 ∧ Ord 𝑢) → (𝑣 = 𝑢 ↔ ¬ (𝑣 ∈ 𝑢 ∨ 𝑢 ∈ 𝑣)))
24	21, 22, 23	syl2an 597	. . . . . 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 3181	. . 3 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ∀𝑣 ∈ ω ∀𝑢 ∈ ω ((𝐹‘𝑣) = (𝐹‘𝑢) → 𝑣 = 𝑢))
28		frfnom 8368	. . . . . 6 ⊢ (rec(𝐺, ∅) ↾ ω) Fn ω
29		fneq1 6584	. . . . . 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 6895	. . . . . . . 8 ⊢ (𝐹 Fn ω → (𝑢 ∈ ran 𝐹 ↔ ∃𝑣 ∈ ω (𝐹‘𝑣) = 𝑢))
32		inf3lem.4	. . . . . . . . . . . 12 ⊢ 𝐵 ∈ V
33	1, 2, 4, 32	inf3lemd 9542	. . . . . . . . . . 11 ⊢ (𝑣 ∈ ω → (𝐹‘𝑣) ⊆ 𝑥)
34		fvex 6848	. . . . . . . . . . . 12 ⊢ (𝐹‘𝑣) ∈ V
35	34	elpw 4546	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑣) ∈ 𝒫 𝑥 ↔ (𝐹‘𝑣) ⊆ 𝑥)
36	33, 35	sylibr 234	. . . . . . . . . 10 ⊢ (𝑣 ∈ ω → (𝐹‘𝑣) ∈ 𝒫 𝑥)
37		eleq1 2825	. . . . . . . . . 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 3132	. . . . . . . 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 3928	. . . . . 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 6497	. . . 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 7203	. 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 848   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  {crab 3390  Vcvv 3430   ∩ cin 3889   ⊆ wss 3890   ⊊ wpss 3891  ∅c0 4274  𝒫 cpw 4542  ∪ cuni 4851   ↦ cmpt 5167  ran crn 5626   ↾ cres 5627  Ord word 6317   Fn wfn 6488  ⟶wf 6489  –1-1→wf1 6490  ‘cfv 6493  ωcom 7811  reccrdg 8342

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5371  ax-un 7683  ax-reg 9501

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7364  df-om 7812  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343

This theorem is referenced by:  inf3lem7  9549  dominf  10361  dominfac  10490