Theorem dfac5lem5 9814

Description: Lemma for dfac5 9815. (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 9813	. 2 ⊢ (𝜑 → ∃𝑦∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))
5		simpr 484	. . . . . . . . . 10 ⊢ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ ℎ) → 𝑤 ∈ ℎ)
6	5	a1i 11	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) → ((𝑤 ≠ ∅ ∧ 𝑤 ∈ ℎ) → 𝑤 ∈ ℎ))
7		ineq1 4136	. . . . . . . . . . . . 13 ⊢ (𝑧 = ({𝑤} × 𝑤) → (𝑧 ∩ 𝑦) = (({𝑤} × 𝑤) ∩ 𝑦))
8	7	eleq2d 2824	. . . . . . . . . . . 12 ⊢ (𝑧 = ({𝑤} × 𝑤) → (𝑣 ∈ (𝑧 ∩ 𝑦) ↔ 𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦)))
9	8	eubidv 2586	. . . . . . . . . . 11 ⊢ (𝑧 = ({𝑤} × 𝑤) → (∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) ↔ ∃!𝑣 𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦)))
10	9	rspccv 3549	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) → (({𝑤} × 𝑤) ∈ 𝐴 → ∃!𝑣 𝑣 ∈ (({𝑤} × 𝑤) ∩ 𝑦)))
11	1	dfac5lem3 9812	. . . . . . . . . 10 ⊢ (({𝑤} × 𝑤) ∈ 𝐴 ↔ (𝑤 ≠ ∅ ∧ 𝑤 ∈ ℎ))
12		dfac5lem1 9810	. . . . . . . . . 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 512	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) → ((𝑤 ≠ ∅ ∧ 𝑤 ∈ ℎ) → (𝑤 ∈ ℎ ∧ ∃!𝑔(𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))))
15	2	eleq2i 2830	. . . . . . . . . . 11 ⊢ (⟨𝑤, 𝑔⟩ ∈ 𝐵 ↔ ⟨𝑤, 𝑔⟩ ∈ (∪ 𝐴 ∩ 𝑦))
16		elin 3899	. . . . . . . . . . 11 ⊢ (⟨𝑤, 𝑔⟩ ∈ (∪ 𝐴 ∩ 𝑦) ↔ (⟨𝑤, 𝑔⟩ ∈ ∪ 𝐴 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦))
17	1	dfac5lem2 9811	. . . . . . . . . . . . 13 ⊢ (⟨𝑤, 𝑔⟩ ∈ ∪ 𝐴 ↔ (𝑤 ∈ ℎ ∧ 𝑔 ∈ 𝑤))
18	17	anbi1i 623	. . . . . . . . . . . 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 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 2585	. . . . . . . . 9 ⊢ (∃!𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵 ↔ ∃!𝑔(𝑤 ∈ ℎ ∧ (𝑔 ∈ 𝑤 ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑦)))
23		euanv 2626	. . . . . . . . 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	syl6ib 250	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) → ((𝑤 ≠ ∅ ∧ 𝑤 ∈ ℎ) → ∃!𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵))
26		euex 2577	. . . . . . . 8 ⊢ (∃!𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵 → ∃𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵)
27		nfeu1 2588	. . . . . . . . . 10 ⊢ Ⅎ𝑔∃!𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵
28		nfv 1918	. . . . . . . . . 10 ⊢ Ⅎ𝑔(𝐵‘𝑤) ∈ 𝑤
29	27, 28	nfim 1900	. . . . . . . . 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 6779	. . . . . . . . . . . . 13 ⊢ ((⟨𝑤, 𝑔⟩ ∈ 𝐵 ∧ ∃!𝑔⟨𝑤, 𝑔⟩ ∈ 𝐵) → (𝐵‘𝑤) = 𝑔)
33	32	eleq1d 2823	. . . . . . . . . . . 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 2213	. . . . . . . 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 3106	. . . 4 ⊢ (∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) → ∀𝑤 ∈ ℎ (𝑤 ≠ ∅ → (𝐵‘𝑤) ∈ 𝑤))
42		vex 3426	. . . . . . 7 ⊢ 𝑦 ∈ V
43	42	inex2 5237	. . . . . 6 ⊢ (∪ 𝐴 ∩ 𝑦) ∈ V
44	2, 43	eqeltri 2835	. . . . 5 ⊢ 𝐵 ∈ V
45		fveq1 6755	. . . . . . . 8 ⊢ (𝑓 = 𝐵 → (𝑓‘𝑤) = (𝐵‘𝑤))
46	45	eleq1d 2823	. . . . . . 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 3120	. . . . 5 ⊢ (𝑓 = 𝐵 → (∀𝑤 ∈ ℎ (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ↔ ∀𝑤 ∈ ℎ (𝑤 ≠ ∅ → (𝐵‘𝑤) ∈ 𝑤)))
49	44, 48	spcev 3535	. . . 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 1934	. 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 395  ∀wal 1537   = wceq 1539  ∃wex 1783   ∈ wcel 2108  ∃!weu 2568  {cab 2715   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ∩ cin 3882  ∅c0 4253  {csn 4558  ⟨cop 4564  ∪ cuni 4836   × cxp 5578  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-cnv 5588  df-dm 5590  df-rn 5591  df-iota 6376  df-fv 6426

This theorem is referenced by:  dfac5  9815