Theorem dfac5lem5 10018

Description: Lemma for dfac5 10020. (Contributed by NM, 12-Apr-2004.)

Hypotheses

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

Assertion

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

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

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 ⊢ (𝜑 ↔ ∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)))
3	1, 2	dfac5lem4 10017	. 2 ⊢ (𝜑 → ∃𝑦∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))
4		simpr 484	. . . . . . . . . 10 ⊢ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ ℎ) → 𝑤 ∈ ℎ)
5	4	a1i 11	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) → ((𝑤 ≠ ∅ ∧ 𝑤 ∈ ℎ) → 𝑤 ∈ ℎ))
6		ineq1 4160	. . . . . . . . . . . . 13 ⊢ (𝑧 = ({𝑤} × 𝑤) → (𝑧 ∩ 𝑦) = (({𝑤} × 𝑤) ∩ 𝑦))
7	6	eleq2d 2817	. . . . . . . . . . . 12 ⊢ (𝑧 = ({𝑤} × 𝑤) → (𝑣 ∈ (𝑧 ∩ 𝑦) ↔ 𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦)))
8	7	eubidv 2581	. . . . . . . . . . 11 ⊢ (𝑧 = ({𝑤} × 𝑤) → (∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) ↔ ∃!𝑣 𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦)))
9	8	rspccv 3569	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) → (({𝑤} × 𝑤) ∈ 𝐴 → ∃!𝑣 𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦)))
10	1	dfac5lem3 10016	. . . . . . . . . 10 ⊢ (({𝑤} × 𝑤) ∈ 𝐴 ↔ (𝑤 ≠ ∅ ∧ 𝑤 ∈ ℎ))
11		dfac5lem1 10014	. . . . . . . . . 10 ⊢ (∃!𝑣 𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦) ↔ ∃!𝑔(𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))
12	9, 10, 11	3imtr3g 295	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) → ((𝑤 ≠ ∅ ∧ 𝑤 ∈ ℎ) → ∃!𝑔(𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
13	5, 12	jcad 512	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) → ((𝑤 ≠ ∅ ∧ 𝑤 ∈ ℎ) → (𝑤 ∈ ℎ ∧ ∃!𝑔(𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))))
14		dfac5lem.3	. . . . . . . . . . . 12 ⊢ 𝐵 = (∪ 𝐴 ∩ 𝑦)
15	14	eleq2i 2823	. . . . . . . . . . 11 ⊢ (⟨𝑤, 𝑔⟩ ∈ 𝐵 ↔ ⟨𝑤, 𝑔⟩ ∈ (∪ 𝐴 ∩ 𝑦))
16		elin 3913	. . . . . . . . . . 11 ⊢ (⟨𝑤, 𝑔⟩ ∈ (∪ 𝐴 ∩ 𝑦) ↔ (⟨𝑤, 𝑔⟩ ∈ ∪ 𝐴 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))
17	1	dfac5lem2 10015	. . . . . . . . . . . . 13 ⊢ (⟨𝑤, 𝑔⟩ ∈ ∪ 𝐴 ↔ (𝑤 ∈ ℎ ∧ 𝑔 ∈ 𝑤))
18	17	anbi1i 624	. . . . . . . . . . . 12 ⊢ ((⟨𝑤, 𝑔⟩ ∈ ∪ 𝐴 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦) ↔ ((𝑤 ∈ ℎ ∧ 𝑔 ∈ 𝑤) ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))
19		anass 468	. . . . . . . . . . . 12 ⊢ (((𝑤 ∈ ℎ ∧ 𝑔 ∈ 𝑤) ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦) ↔ (𝑤 ∈ ℎ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
20	18, 19	bitri 275	. . . . . . . . . . 11 ⊢ ((⟨𝑤, 𝑔⟩ ∈ ∪ 𝐴 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦) ↔ (𝑤 ∈ ℎ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
21	15, 16, 20	3bitri 297	. . . . . . . . . 10 ⊢ (⟨𝑤, 𝑔⟩ ∈ 𝐵 ↔ (𝑤 ∈ ℎ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
22	21	eubii 2580	. . . . . . . . 9 ⊢ (∃!𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵 ↔ ∃!𝑔(𝑤 ∈ ℎ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
23		euanv 2619	. . . . . . . . 9 ⊢ (∃!𝑔(𝑤 ∈ ℎ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)) ↔ (𝑤 ∈ ℎ ∧ ∃!𝑔(𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
24	22, 23	bitr2i 276	. . . . . . . 8 ⊢ ((𝑤 ∈ ℎ ∧ ∃!𝑔(𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)) ↔ ∃!𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵)
25	13, 24	imbitrdi 251	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) → ((𝑤 ≠ ∅ ∧ 𝑤 ∈ ℎ) → ∃!𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵))
26		euex 2572	. . . . . . . 8 ⊢ (∃!𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵 → ∃𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵)
27		nfeu1 2583	. . . . . . . . . 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 496	. . . . . . . . . . 11 ⊢ (⟨𝑤, 𝑔⟩ ∈ 𝐵 → (𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))
31	30	simpld 494	. . . . . . . . . 10 ⊢ (⟨𝑤, 𝑔⟩ ∈ 𝐵 → 𝑔 ∈ 𝑤)
32		tz6.12 6846	. . . . . . . . . . . . 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 479	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ 𝑤 ∧ (⟨𝑤, 𝑔⟩ ∈ 𝐵 ∧ ∃!𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵)) → (𝐵‘𝑤) ∈ 𝑤)
35	34	exp32 420	. . . . . . . . . 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 2220	. . . . . . . 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 416	. . . . 5 ⊢ (∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) → (𝑤 ∈ ℎ → (𝑤 ≠ ∅ → (𝐵‘𝑤) ∈ 𝑤)))
41	40	ralrimiv 3123	. . . 4 ⊢ (∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) → ∀𝑤 ∈ ℎ (𝑤 ≠ ∅ → (𝐵‘𝑤) ∈ 𝑤))
42		vex 3440	. . . . . . 7 ⊢ 𝑦 ∈ V
43	42	inex2 5254	. . . . . 6 ⊢ (∪ 𝐴 ∩ 𝑦) ∈ V
44	14, 43	eqeltri 2827	. . . . 5 ⊢ 𝐵 ∈ V
45		fveq1 6821	. . . . . . . 8 ⊢ (𝑓 = 𝐵 → (𝑓‘𝑤) = (𝐵‘𝑤))
46	45	eleq1d 2816	. . . . . . 7 ⊢ (𝑓 = 𝐵 → ((𝑓‘𝑤) ∈ 𝑤 ↔ (𝐵‘𝑤) ∈ 𝑤))
47	46	imbi2d 340	. . . . . 6 ⊢ (𝑓 = 𝐵 → ((𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ↔ (𝑤 ≠ ∅ → (𝐵‘𝑤) ∈ 𝑤)))
48	47	ralbidv 3155	. . . . 5 ⊢ (𝑓 = 𝐵 → (∀𝑤 ∈ ℎ (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ↔ ∀𝑤 ∈ ℎ (𝑤 ≠ ∅ → (𝐵‘𝑤) ∈ 𝑤)))
49	44, 48	spcev 3556	. . . 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	3, 51	syl 17	1 ⊢ (𝜑 → ∃𝑓∀𝑤 ∈ ℎ (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2111  ∃!weu 2563  {cab 2709   ≠ wne 2928  ∀wral 3047  ∃wrex 3056  Vcvv 3436   ∩ cin 3896  ∅c0 4280  {csn 4573  ⟨cop 4579  ∪ cuni 4856   × cxp 5612  ‘cfv 6481

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-xp 5620  df-rel 5621  df-cnv 5622  df-dm 5624  df-rn 5625  df-iota 6437  df-fv 6489

This theorem is referenced by:  dfac5  10020