Theorem dfac5lem5 10124

Description: Lemma for dfac5 10125. (Contributed by NM, 12-Apr-2004.)

Hypotheses

Ref	Expression
dfac5lem.1	⊢ 𝐴 = {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 ∈ ℎ 𝑢 = ({𝑡} × 𝑡))}
dfac5lem.2	⊢ 𝐵 = (∪ 𝐴 ∩ 𝑦)
dfac5lem.3	⊢ (𝜑 ↔ ∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)))

Assertion

Ref	Expression
dfac5lem5	⊢ (𝜑 → ∃𝑓∀𝑤 ∈ ℎ (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤))

Distinct variable groups:   𝑥,𝑓,𝑧,𝑦,𝑤,𝑣,𝑢,𝑡,ℎ   𝑧,𝐵,𝑤,𝑓   𝑥,𝐴,𝑦,𝑧,𝑤

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,𝑓,ℎ)   𝐴(𝑣,𝑢,𝑡,𝑓,ℎ)   𝐵(𝑥,𝑦,𝑣,𝑢,𝑡,ℎ)

Proof of Theorem dfac5lem5

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfac5lem.1	. . 3 ⊢ 𝐴 = {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 ∈ ℎ 𝑢 = ({𝑡} × 𝑡))}
2		dfac5lem.2	. . 3 ⊢ 𝐵 = (∪ 𝐴 ∩ 𝑦)
3		dfac5lem.3	. . 3 ⊢ (𝜑 ↔ ∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)))
4	1, 2, 3	dfac5lem4 10123	. 2 ⊢ (𝜑 → ∃𝑦∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))
5		simpr 483	. . . . . . . . . 10 ⊢ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ ℎ) → 𝑤 ∈ ℎ)
6	5	a1i 11	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) → ((𝑤 ≠ ∅ ∧ 𝑤 ∈ ℎ) → 𝑤 ∈ ℎ))
7		ineq1 4204	. . . . . . . . . . . . 13 ⊢ (𝑧 = ({𝑤} × 𝑤) → (𝑧 ∩ 𝑦) = (({𝑤} × 𝑤) ∩ 𝑦))
8	7	eleq2d 2817	. . . . . . . . . . . 12 ⊢ (𝑧 = ({𝑤} × 𝑤) → (𝑣 ∈ (𝑧 ∩ 𝑦) ↔ 𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦)))
9	8	eubidv 2578	. . . . . . . . . . 11 ⊢ (𝑧 = ({𝑤} × 𝑤) → (∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) ↔ ∃!𝑣 𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦)))
10	9	rspccv 3608	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) → (({𝑤} × 𝑤) ∈ 𝐴 → ∃!𝑣 𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦)))
11	1	dfac5lem3 10122	. . . . . . . . . 10 ⊢ (({𝑤} × 𝑤) ∈ 𝐴 ↔ (𝑤 ≠ ∅ ∧ 𝑤 ∈ ℎ))
12		dfac5lem1 10120	. . . . . . . . . 10 ⊢ (∃!𝑣 𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦) ↔ ∃!𝑔(𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))
13	10, 11, 12	3imtr3g 294	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) → ((𝑤 ≠ ∅ ∧ 𝑤 ∈ ℎ) → ∃!𝑔(𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
14	6, 13	jcad 511	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) → ((𝑤 ≠ ∅ ∧ 𝑤 ∈ ℎ) → (𝑤 ∈ ℎ ∧ ∃!𝑔(𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))))
15	2	eleq2i 2823	. . . . . . . . . . 11 ⊢ (⟨𝑤, 𝑔⟩ ∈ 𝐵 ↔ ⟨𝑤, 𝑔⟩ ∈ (∪ 𝐴 ∩ 𝑦))
16		elin 3963	. . . . . . . . . . 11 ⊢ (⟨𝑤, 𝑔⟩ ∈ (∪ 𝐴 ∩ 𝑦) ↔ (⟨𝑤, 𝑔⟩ ∈ ∪ 𝐴 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))
17	1	dfac5lem2 10121	. . . . . . . . . . . . 13 ⊢ (⟨𝑤, 𝑔⟩ ∈ ∪ 𝐴 ↔ (𝑤 ∈ ℎ ∧ 𝑔 ∈ 𝑤))
18	17	anbi1i 622	. . . . . . . . . . . 12 ⊢ ((⟨𝑤, 𝑔⟩ ∈ ∪ 𝐴 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦) ↔ ((𝑤 ∈ ℎ ∧ 𝑔 ∈ 𝑤) ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))
19		anass 467	. . . . . . . . . . . 12 ⊢ (((𝑤 ∈ ℎ ∧ 𝑔 ∈ 𝑤) ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦) ↔ (𝑤 ∈ ℎ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
20	18, 19	bitri 274	. . . . . . . . . . 11 ⊢ ((⟨𝑤, 𝑔⟩ ∈ ∪ 𝐴 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦) ↔ (𝑤 ∈ ℎ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
21	15, 16, 20	3bitri 296	. . . . . . . . . 10 ⊢ (⟨𝑤, 𝑔⟩ ∈ 𝐵 ↔ (𝑤 ∈ ℎ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
22	21	eubii 2577	. . . . . . . . 9 ⊢ (∃!𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵 ↔ ∃!𝑔(𝑤 ∈ ℎ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
23		euanv 2618	. . . . . . . . 9 ⊢ (∃!𝑔(𝑤 ∈ ℎ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)) ↔ (𝑤 ∈ ℎ ∧ ∃!𝑔(𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
24	22, 23	bitr2i 275	. . . . . . . 8 ⊢ ((𝑤 ∈ ℎ ∧ ∃!𝑔(𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)) ↔ ∃!𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵)
25	14, 24	imbitrdi 250	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) → ((𝑤 ≠ ∅ ∧ 𝑤 ∈ ℎ) → ∃!𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵))
26		euex 2569	. . . . . . . 8 ⊢ (∃!𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵 → ∃𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵)
27		nfeu1 2580	. . . . . . . . . 10 ⊢ Ⅎ𝑔∃!𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵
28		nfv 1915	. . . . . . . . . 10 ⊢ Ⅎ𝑔(𝐵‘𝑤) ∈ 𝑤
29	27, 28	nfim 1897	. . . . . . . . 9 ⊢ Ⅎ𝑔(∃!𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵 → (𝐵‘𝑤) ∈ 𝑤)
30	21	simprbi 495	. . . . . . . . . . 11 ⊢ (⟨𝑤, 𝑔⟩ ∈ 𝐵 → (𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))
31	30	simpld 493	. . . . . . . . . 10 ⊢ (⟨𝑤, 𝑔⟩ ∈ 𝐵 → 𝑔 ∈ 𝑤)
32		tz6.12 6915	. . . . . . . . . . . . 13 ⊢ ((⟨𝑤, 𝑔⟩ ∈ 𝐵 ∧ ∃!𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵) → (𝐵‘𝑤) = 𝑔)
33	32	eleq1d 2816	. . . . . . . . . . . 12 ⊢ ((⟨𝑤, 𝑔⟩ ∈ 𝐵 ∧ ∃!𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵) → ((𝐵‘𝑤) ∈ 𝑤 ↔ 𝑔 ∈ 𝑤))
34	33	biimparc 478	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ 𝑤 ∧ (⟨𝑤, 𝑔⟩ ∈ 𝐵 ∧ ∃!𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵)) → (𝐵‘𝑤) ∈ 𝑤)
35	34	exp32 419	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝑤 → (⟨𝑤, 𝑔⟩ ∈ 𝐵 → (∃!𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵 → (𝐵‘𝑤) ∈ 𝑤)))
36	31, 35	mpcom 38	. . . . . . . . 9 ⊢ (⟨𝑤, 𝑔⟩ ∈ 𝐵 → (∃!𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵 → (𝐵‘𝑤) ∈ 𝑤))
37	29, 36	exlimi 2208	. . . . . . . 8 ⊢ (∃𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵 → (∃!𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵 → (𝐵‘𝑤) ∈ 𝑤))
38	26, 37	mpcom 38	. . . . . . 7 ⊢ (∃!𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵 → (𝐵‘𝑤) ∈ 𝑤)
39	25, 38	syl6 35	. . . . . 6 ⊢ (∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) → ((𝑤 ≠ ∅ ∧ 𝑤 ∈ ℎ) → (𝐵‘𝑤) ∈ 𝑤))
40	39	expcomd 415	. . . . 5 ⊢ (∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) → (𝑤 ∈ ℎ → (𝑤 ≠ ∅ → (𝐵‘𝑤) ∈ 𝑤)))
41	40	ralrimiv 3143	. . . 4 ⊢ (∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) → ∀𝑤 ∈ ℎ (𝑤 ≠ ∅ → (𝐵‘𝑤) ∈ 𝑤))
42		vex 3476	. . . . . . 7 ⊢ 𝑦 ∈ V
43	42	inex2 5317	. . . . . 6 ⊢ (∪ 𝐴 ∩ 𝑦) ∈ V
44	2, 43	eqeltri 2827	. . . . 5 ⊢ 𝐵 ∈ V
45		fveq1 6889	. . . . . . . 8 ⊢ (𝑓 = 𝐵 → (𝑓‘𝑤) = (𝐵‘𝑤))
46	45	eleq1d 2816	. . . . . . 7 ⊢ (𝑓 = 𝐵 → ((𝑓‘𝑤) ∈ 𝑤 ↔ (𝐵‘𝑤) ∈ 𝑤))
47	46	imbi2d 339	. . . . . 6 ⊢ (𝑓 = 𝐵 → ((𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ↔ (𝑤 ≠ ∅ → (𝐵‘𝑤) ∈ 𝑤)))
48	47	ralbidv 3175	. . . . 5 ⊢ (𝑓 = 𝐵 → (∀𝑤 ∈ ℎ (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ↔ ∀𝑤 ∈ ℎ (𝑤 ≠ ∅ → (𝐵‘𝑤) ∈ 𝑤)))
49	44, 48	spcev 3595	. . . 4 ⊢ (∀𝑤 ∈ ℎ (𝑤 ≠ ∅ → (𝐵‘𝑤) ∈ 𝑤) → ∃𝑓∀𝑤 ∈ ℎ (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤))
50	41, 49	syl 17	. . 3 ⊢ (∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) → ∃𝑓∀𝑤 ∈ ℎ (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤))
51	50	exlimiv 1931	. 2 ⊢ (∃𝑦∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) → ∃𝑓∀𝑤 ∈ ℎ (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤))
52	4, 51	syl 17	1 ⊢ (𝜑 → ∃𝑓∀𝑤 ∈ ℎ (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394  ∀wal 1537   = wceq 1539  ∃wex 1779   ∈ wcel 2104  ∃!weu 2560  {cab 2707   ≠ wne 2938  ∀wral 3059  ∃wrex 3068  Vcvv 3472   ∩ cin 3946  ∅c0 4321  {csn 4627  ⟨cop 4633  ∪ cuni 4907   × cxp 5673  ‘cfv 6542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-cnv 5683  df-dm 5685  df-rn 5686  df-iota 6494  df-fv 6550

This theorem is referenced by:  dfac5  10125